{"id":4546,"date":"2016-05-16T11:49:00","date_gmt":"2016-05-16T06:19:00","guid":{"rendered":"http:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-3\/"},"modified":"2018-05-04T14:57:51","modified_gmt":"2018-05-04T09:27:51","slug":"ncert-solutions-class-12-maths-exercise-6-3","status":"publish","type":"post","link":"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-3\/","title":{"rendered":"NCERT Solutions class 12 Maths Exercise 6.3"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_76 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 eztoc-toggle-hide-by-default' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-3\/#NCERT_Solutions_for_Class_12_Maths_Application_of_Derivatives\" >NCERT Solutions for Class 12 Maths\u00a0Application of Derivatives\u00a0<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-3\/#NCERT_Solutions_class_12_Maths_Exercise_63\" >NCERT Solutions class 12 Maths Exercise 6.3<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-3\/#2_Find_the_slope_of_tangent_to_the_curve_at\" >2. Find the slope of tangent to the curve \u00a0at \u00a0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-3\/#NCERT_Solutions_class_12_Maths_Exercise_63-2\" >NCERT Solutions class 12 Maths Exercise 6.3<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-3\/#3_Find_the_slope_of_tangent_to_the_curve_at_the_given_point_whose_coordinate_is_2\" >3. Find the slope of tangent to the curve \u00a0at the given point whose coordinate is 2.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-3\/#NCERT_Solutions_class_12_Maths_Exercise_63-3\" >NCERT Solutions class 12 Maths Exercise 6.3<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-3\/#4_Find_the_slope_of_tangent_to_the_curve_at_the_given_point_whose_coordinate_is_3\" >4. Find the slope of tangent to the curve \u00a0at the given point whose coordinate is 3.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-3\/#NCERT_Solutions_class_12_Maths_Exercise_63-4\" >NCERT Solutions class 12 Maths Exercise 6.3<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-3\/#5_Find_the_slope_of_the_normal_to_the_curve_at\" >5. Find the slope of the normal to the curve \u00a0at \u00a0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-3\/#NCERT_Solutions_class_12_Maths_Exercise_63-5\" >NCERT Solutions class 12 Maths Exercise 6.3<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-3\/#6_Find_the_slope_of_the_normal_to_the_curve_at\" >6. Find the slope of the normal to the curve \u00a0at \u00a0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-3\/#NCERT_Solutions_class_12_Maths_Exercise_63-6\" >NCERT Solutions class 12 Maths Exercise 6.3<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-3\/#7_Find_the_point_at_which_the_tangent_to_the_curve_is_parallel_to_the_axis\" >7. Find the point at which the tangent to the curve \u00a0is parallel to the axis.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-14\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-3\/#NCERT_Solutions_class_12_Maths_Exercise_63-7\" >NCERT Solutions class 12 Maths Exercise 6.3<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-15\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-3\/#8_Find_the_point_on_the_curve_at_which_the_tangent_is_parallel_to_the_chord_joining_the_points_2_0_and_4_4\" >8. Find the point on the curve \u00a0at which the tangent is parallel to the chord joining the points (2, 0) and (4, 4).<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-16\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-3\/#13_Find_the_points_on_the_curve_at_which_the_tangents_are\" >13. Find the points on the curve \u00a0at which the tangents are:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-17\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-3\/#14_Find_the_equation_of_the_tangents_and_normal_to_the_given_curves_at_the_indicated_points\" >14. Find the equation of the tangents and normal to the given curves at the indicated points:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-18\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-3\/#15_Find_the_equation_of_the_tangent_line_to_curve_which_is\" >15. Find the equation of the tangent line to curve \u00a0which is:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-19\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-3\/#16_Show_that_the_tangents_to_the_curve_at_the_points_where_and_are_parallel\" >16. Show that the tangents to the curve \u00a0at the points where \u00a0and \u00a0are parallel.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-20\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-3\/#17_Find_the_points_on_the_curve_at_which_the_slope_of_the_tangent_is_equal_to_the_coordinate_of_the_point\" >17. Find the points on the curve \u00a0at which the slope of the tangent is equal to the coordinate of the point.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-21\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-3\/#18_For_the_curve_find_all_point_at_which_the_tangent_passes_through_the_origin\" >18. For the curve \u00a0find all point at which the tangent passes through the origin.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-22\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-3\/#19_Find_the_points_on_the_curve_at_which_the_tangents_are_parallel_to_axis\" >19. Find the points on the curve \u00a0at which the tangents are parallel to axis.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-23\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-3\/#20_Find_the_equation_of_the_normal_at_the_point_for_the_curve\" >20. Find the equation of the normal at the point \u00a0for the curve \u00a0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-24\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-3\/#21_Find_the_equations_of_the_normal_to_the_curve_which_are_parallel_to_the_line\" >21. Find the equations of the normal to the curve \u00a0which are parallel to the line \u00a0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-25\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-3\/#22_Find_the_equation_of_the_tangent_and_normal_to_the_parabola_at_the_point\" >22. Find the equation of the tangent and normal to the parabola \u00a0at the point \u00a0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-26\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-3\/#23_Prove_that_the_curves_and_cut_at_right_angles_if\" >23. Prove that the curves \u00a0and \u00a0cut at right angles if \u00a0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-27\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-3\/#24_Find_the_equation_of_the_tangent_and_normal_to_the_hyperbola_at_the_point\" >24. Find the equation of the tangent and normal to the hyperbola \u00a0at the point \u00a0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-28\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-3\/#NCERT_Solutions_class_12_Maths_Exercise_63-8\" >NCERT Solutions class 12 Maths Exercise 6.3<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-29\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-3\/#25_Find_the_equation_of_the_tangent_to_the_curve_which_is_parallel_to_the_line\" >25. Find the equation of the tangent to the curve \u00a0which is parallel to the line \u00a0<\/a><ul class='ez-toc-list-level-4' ><li class='ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-30\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-3\/#According_to_the_question_Parallel_lines_have_same_slope\" >According to the question, \u00a0[Parallel lines have same slope]<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-31\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-3\/#26_The_slope_of_the_normal_to_the_curve_at_is\" >26. The slope of the normal to the curve \u00a0at \u00a0is:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-32\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-3\/#27_The_line_is_a_tangent_to_the_curve_at_the_point\" >27. The line \u00a0is a tangent to the curve \u00a0at the point:<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-33\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-3\/#NCERT_Solutions_class_12_Maths_Exercise_63-9\" >NCERT Solutions class 12 Maths Exercise 6.3<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-34\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-3\/#CBSE_App_for_Class_12\" >CBSE App for Class 12<\/a><\/li><\/ul><\/nav><\/div>\n<div class=\"addtoany_share_save_container addtoany_content_top\">\n<div class=\"a2a_kit a2a_kit_size_50 addtoany_list\" data-a2a-url=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-1\/\" data-a2a-title=\"NCERT Solutions class 12 Maths Exercise 6.1\">NCERT Solutions class 12 Maths Exercise 6.3 Class 12 Maths Class book solutions are available in PDF format for free download. These ncert book chapter wise questions and answers are very helpful for CBSE board exam. CBSE recommends NCERT books and most of the questions in CBSE exam are asked from NCERT text books. Class 12 Maths chapter wise NCERT solution for Maths part 1 and Maths part 2 for all the chapters can be downloaded from our website and myCBSEguide mobile app for free.<\/div>\n<\/div>\n<p><strong>Download\u00a0<a href=\"https:\/\/mycbseguide.com\/downloads\/cbse-class-12-mathematics-application-of-derivatives\/1290\/ncert-solutions\/5\/\">NCERT solutions for Applications of Derivatives<\/a>\u00a0as PDF.<\/strong><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/media-mycbseguide.s3.ap-south-1.amazonaws.com\/images\/blog\/12%20Maths%20Book%20pic.jpg\" alt=\"NCERT Solutions class 12 Maths Exercise 6.3\" width=\"187\" height=\"243\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"NCERT_Solutions_for_Class_12_Maths_Application_of_Derivatives\"><\/span><strong>NCERT Solutions for Class 12 Maths\u00a0Application of Derivatives<\/strong><strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><strong>1. Find the slope of tangent to the curve <img decoding=\"async\" style=\"height: 24px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image021.png\" \/>\u00a0at <img decoding=\"async\" style=\"height: 19px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image022.png\" \/>\u00a0<\/strong><\/p>\n<p><strong>Ans. <\/strong>Given: Equation of the curve <img decoding=\"async\" style=\"height: 24px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image021.png\" \/>\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p>Slope of the tangent to the curve = Value of <img decoding=\"async\" style=\"height: 41px; width: 23px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image023.png\" \/>\u00a0at the point <img decoding=\"async\" style=\"height: 27px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image024.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 41px; width: 172px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image026.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0Slope of the tangent at point <img decoding=\"async\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image027.png\" \/>\u00a0to the curve (i)<\/p>\n<p>= <img decoding=\"async\" style=\"height: 30px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image028.png\" \/>\u00a0= 768 \u2013 4 = 764<\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_63\"><\/span>NCERT Solutions class 12 Maths Exercise 6.3<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"2_Find_the_slope_of_tangent_to_the_curve_at\"><\/span><strong>2. Find the slope of tangent to the curve <img decoding=\"async\" style=\"height: 41px; width: 102px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image029.png\" \/>\u00a0at <img decoding=\"async\" style=\"height: 19px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image030.png\" \/>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Ans. <\/strong>Given: Equation of the curve <img decoding=\"async\" style=\"height: 41px; width: 63px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image031.png\" \/>\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 67px; width: 264px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image032.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 52px; width: 104px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image033.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 49px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image034.png\" \/>\u2026\u2026\u2026.(ii)<\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0Slope of the tangent at point <img decoding=\"async\" style=\"height: 19px; width: 44px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image035.png\" \/>\u00a0to the curve (i)<\/p>\n<p>= <img decoding=\"async\" style=\"height: 49px; width: 63px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image036.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 41px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image037.png\" \/><\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_63-2\"><\/span>NCERT Solutions class 12 Maths Exercise 6.3<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"3_Find_the_slope_of_tangent_to_the_curve_at_the_given_point_whose_coordinate_is_2\"><\/span><strong>3. Find the slope of tangent to the curve <img decoding=\"async\" style=\"height: 24px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image038.png\" \/>\u00a0at the given point whose <img decoding=\"async\" style=\"height: 15px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image039.png\" \/>coordinate is 2.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Ans. <\/strong>Given: Equation of the curve <img decoding=\"async\" style=\"height: 24px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image038.png\" \/>\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image040.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0Slope of the tangent at point <img decoding=\"async\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image041.png\" \/>\u00a0to the curve (i)<\/p>\n<p>= <img decoding=\"async\" style=\"height: 30px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image042.png\" \/>\u00a0= 12 \u2013 1 = 11<\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_63-3\"><\/span>NCERT Solutions class 12 Maths Exercise 6.3<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"4_Find_the_slope_of_tangent_to_the_curve_at_the_given_point_whose_coordinate_is_3\"><\/span><strong>4. Find the slope of tangent to the curve <img decoding=\"async\" style=\"height: 24px; width: 96px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image043.png\" \/>\u00a0at the given point whose <img decoding=\"async\" style=\"height: 15px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image039.png\" \/>coordinate is 3.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Ans. <\/strong>Given: Equation of the curve <img decoding=\"async\" style=\"height: 24px; width: 96px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image043.png\" \/>\u2026\u2026\u2026.(i)<\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image044.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0Slope of the tangent at point <img decoding=\"async\" style=\"height: 19px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image045.png\" \/>\u00a0to the curve (i)<\/p>\n<p>= <img decoding=\"async\" style=\"height: 30px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image046.png\" \/>\u00a0= 27 \u2013 3 = 24<\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_63-4\"><\/span>NCERT Solutions class 12 Maths Exercise 6.3<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"5_Find_the_slope_of_the_normal_to_the_curve_at\"><\/span><strong>5. Find the slope of the normal to the curve <img decoding=\"async\" style=\"height: 24px; width: 156px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image047.png\" \/>\u00a0at <img decoding=\"async\" style=\"height: 41px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image048.png\" \/>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Ans. <\/strong>Given: Equations of the curves are <img decoding=\"async\" style=\"height: 24px; width: 156px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image047.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 126px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image049.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 41px; width: 123px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image050.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 41px; width: 147px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image051.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 41px; width: 143px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image052.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 41px; width: 140px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image054.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 41px; width: 131px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image055.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 44px; width: 203px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image056.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 41px; width: 107px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image057.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0Slope of the tangent at <img decoding=\"async\" style=\"height: 41px; width: 42px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image058.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 41px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image059.png\" \/><\/p>\n<p>And Slope of the normal at <img decoding=\"async\" style=\"height: 41px; width: 42px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image058.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 41px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image060.png\" \/>\u00a0= 1<\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_63-5\"><\/span>NCERT Solutions class 12 Maths Exercise 6.3<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"6_Find_the_slope_of_the_normal_to_the_curve_at\"><\/span><strong>6. Find the slope of the normal to the curve <img decoding=\"async\" style=\"height: 24px; width: 168px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image061.png\" \/>\u00a0at <img decoding=\"async\" style=\"height: 41px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image062.png\" \/>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Ans. <\/strong>Given: Equations of the curves are <img decoding=\"async\" style=\"height: 19px; width: 90px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image063.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 24px; width: 79px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image064.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 107px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image065.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 41px; width: 125px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image066.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 41px; width: 92px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image067.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 41px; width: 259px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image068.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 197px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image069.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 41px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image070.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0Slope of the tangent at <img decoding=\"async\" style=\"height: 41px; width: 42px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image071.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 41px; width: 92px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image072.png\" \/><\/p>\n<p>And Slope of the normal at <img decoding=\"async\" style=\"height: 41px; width: 42px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image071.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 41px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image073.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 41px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image074.png\" \/><\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_63-6\"><\/span>NCERT Solutions class 12 Maths Exercise 6.3<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"7_Find_the_point_at_which_the_tangent_to_the_curve_is_parallel_to_the_axis\"><\/span><strong>7. Find the point at which the tangent to the curve <img decoding=\"async\" style=\"height: 24px; width: 133px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image075.png\" \/>\u00a0is parallel to the <img decoding=\"async\" style=\"height: 15px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image039.png\" \/>axis.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Ans. <\/strong>Given: Equation of the curve \u00a0<img decoding=\"async\" style=\"height: 24px; width: 133px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image075.png\" \/>\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 41px; width: 113px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image076.png\" \/><\/p>\n<p>Since, the tangent is parallel to the <img decoding=\"async\" style=\"height: 15px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image039.png\" \/>axis, i.e., <img decoding=\"async\" style=\"height: 41px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image077.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 102px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image078.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 95px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image079.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 112px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image080.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image081.png\" \/><\/p>\n<p>From eq. (i), when <img decoding=\"async\" style=\"height: 21px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image082.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 21px; width: 169px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image083.png\" \/><\/p>\n<p>when <img decoding=\"async\" style=\"height: 21px; width: 48px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image084.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 141px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image085.png\" \/><\/p>\n<p>Therefore, the required points are <img decoding=\"async\" style=\"height: 27px; width: 55px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image086.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 27px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image087.png\" \/><\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_63-7\"><\/span>NCERT Solutions class 12 Maths Exercise 6.3<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"8_Find_the_point_on_the_curve_at_which_the_tangent_is_parallel_to_the_chord_joining_the_points_2_0_and_4_4\"><\/span><strong>8. Find the point on the curve <img decoding=\"async\" style=\"height: 30px; width: 79px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image088.png\" \/>\u00a0at which the tangent is parallel to the chord joining the points (2, 0) and (4, 4).<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Ans. <\/strong>Let the given points are A (2, 0) and B (4, 4).<\/p>\n<p>Slope of the chord AB = <img decoding=\"async\" style=\"height: 41px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image089.png\" \/><img decoding=\"async\" style=\"height: 51px; width: 105px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image090.png\" \/><\/p>\n<p>Equation of the curve is <img decoding=\"async\" style=\"height: 30px; width: 79px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image088.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0 Slope of the tangent at <img decoding=\"async\" style=\"height: 27px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image024.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 41px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image091.png\" \/><\/p>\n<p>If the tangent is parallel to the chord AB, then Slope of tangent = Slope of chord<\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 27px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image092.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 19px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image045.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 30px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image093.png\" \/><\/p>\n<p>Therefore, the required point is (3, 1).<\/p>\n<hr \/>\n<p><strong>9. Find the point on the curve <img decoding=\"async\" style=\"height: 24px; width: 101px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image094.png\" \/>\u00a0at which the tangent is <img decoding=\"async\" style=\"height: 21px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image095.png\" \/>\u00a0<\/strong><\/p>\n<p><strong>Ans. <\/strong>Given: Equation of the curve <img decoding=\"async\" style=\"height: 24px; width: 101px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image094.png\" \/>\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p>Equation of the tangent <img decoding=\"async\" style=\"height: 21px; width: 66px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image096.png\" \/>\u2026\u2026\u2026.(ii)<\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image097.png\" \/><\/p>\n<p>From eq. (i), <img decoding=\"async\" style=\"height: 41px; width: 88px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image098.png\" \/><\/p>\n<p>= Slope of the tangent at <img decoding=\"async\" style=\"height: 27px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image024.png\" \/><\/p>\n<p>But from eq. (ii), the slope of tangent = <img decoding=\"async\" style=\"height: 41px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image099.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image100.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 44px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image101.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 19px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image102.png\" \/><\/p>\n<p>From eq. (i), when <img decoding=\"async\" style=\"height: 21px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image103.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 121px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image104.png\" \/><\/p>\n<p>And when <img decoding=\"async\" style=\"height: 21px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image105.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 129px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image106.png\" \/><\/p>\n<p>Since <img decoding=\"async\" style=\"height: 27px; width: 54px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image107.png\" \/>\u00a0does not satisfy eq. (ii), therefore the required point is <img decoding=\"async\" style=\"height: 27px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image108.png\" \/><\/p>\n<hr \/>\n<p><strong>10. Find the equation of all lines having slope <img decoding=\"async\" style=\"height: 17px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image109.png\" \/>\u00a0that are tangents to the curve <img decoding=\"async\" style=\"height: 41px; width: 101px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image110.png\" \/>\u00a0<\/strong><\/p>\n<p><strong>Ans. <\/strong>Given: Equation of the curve <img decoding=\"async\" style=\"height: 41px; width: 126px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image111.png\" \/>\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 181px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image112.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 49px; width: 54px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image113.png\" \/>\u00a0= Slope of the tangent at <img decoding=\"async\" style=\"height: 27px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image024.png\" \/><\/p>\n<p>But according to question, slope = <img decoding=\"async\" style=\"height: 17px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image109.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 49px; width: 54px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image113.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 17px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image109.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 30px; width: 72px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image114.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 19px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image115.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 19px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image116.png\" \/>\u00a0or <img decoding=\"async\" style=\"height: 19px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image117.png\" \/><\/p>\n<p>From eq. (i), when <img decoding=\"async\" style=\"height: 21px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image103.png\" \/><img decoding=\"async\" style=\"height: 41px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image118.png\" \/><\/p>\n<p>And when <img decoding=\"async\" style=\"height: 21px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image119.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image120.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0 Points of contact are (2, 1) and <img decoding=\"async\" style=\"height: 27px; width: 51px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image121.png\" \/><\/p>\n<p>And Equation of two tangents are\u00a0 <img decoding=\"async\" style=\"height: 27px; width: 109px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image122.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 21px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image123.png\" \/>\u00a0and<\/p>\n<p><img decoding=\"async\" style=\"height: 27px; width: 132px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image124.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 21px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image125.png\" \/><\/p>\n<hr \/>\n<p><strong>11. Find the equations of all lines having slope 2 which are tangents to the curve <img decoding=\"async\" style=\"height: 41px; width: 105px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image126.png\" \/><\/strong><\/p>\n<p><strong>Ans. <\/strong>Given: Equation of the curve <img decoding=\"async\" style=\"height: 41px; width: 129px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image127.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 41px; width: 121px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image128.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 49px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image129.png\" \/>\u00a0= Slope of the tangent at <img decoding=\"async\" style=\"height: 27px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image024.png\" \/><\/p>\n<p>But according to question, slope = <img decoding=\"async\" style=\"height: 17px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image130.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 49px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image129.png\" \/>\u00a0= 2<\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 41px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image131.png\" \/>which is not possible.<\/p>\n<p>Hence, there is no tangent to the given curve having slope 2.<\/p>\n<hr \/>\n<p><strong>12. Find the equations of all lines having slope 0 which are tangents to the curve <img decoding=\"async\" style=\"height: 41px; width: 104px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image132.png\" \/><\/strong><\/p>\n<p><strong>Ans. <\/strong>Given: Equation of the curve <img decoding=\"async\" style=\"height: 41px; width: 101px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image133.png\" \/>\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 163px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image134.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 31px; width: 163px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image135.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 56px; width: 92px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image136.png\" \/><\/p>\n<p>But according to question, slope = 0<\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 56px; width: 92px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image136.png\" \/>\u00a0= 0<\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image137.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 19px; width: 35px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image138.png\" \/><\/p>\n<p>From eq. (i), <img decoding=\"async\" style=\"height: 41px; width: 108px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image139.png\" \/><\/p>\n<p>Therefore, the point on the curve which tangent has slope 0 is <img decoding=\"async\" style=\"height: 45px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image140.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0Equation of the tangent is <img decoding=\"async\" style=\"height: 41px; width: 104px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image141.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 63px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image142.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image143.png\" \/><\/p>\n<hr \/>\n<h3><span class=\"ez-toc-section\" id=\"13_Find_the_points_on_the_curve_at_which_the_tangents_are\"><\/span><strong>13. Find the points on the curve <img decoding=\"async\" style=\"height: 44px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image144.png\" \/>\u00a0at which the tangents are:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>(i) parallel to <img decoding=\"async\" style=\"height: 15px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image039.png\" \/>axis\u00a0 <\/strong><\/p>\n<p><strong>(ii) parallel to <img decoding=\"async\" style=\"height: 17px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image145.png\" \/>axis<\/strong><\/p>\n<p><strong>Ans. <\/strong>Given: Equation of the curve <img decoding=\"async\" style=\"height: 44px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image144.png\" \/>\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 41px; width: 102px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image146.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image147.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 97px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image148.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 44px; width: 126px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image149.png\" \/> \u2026\u2026\u2026.(ii)<\/p>\n<p><strong>(i) <\/strong>If tangent is parallel to <img decoding=\"async\" style=\"height: 15px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image039.png\" \/>axis, then Slope of tangent = 0<\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 41px; width: 23px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image023.png\" \/>\u00a0= 0<\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 44px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image150.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image151.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0 From eq. (i), <img decoding=\"async\" style=\"height: 44px; width: 44px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image152.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 24px; width: 51px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image153.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image154.png\" \/><\/p>\n<p>Therefore, the points on curve (i) where tangents are parallel to <img decoding=\"async\" style=\"height: 15px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image039.png\" \/>axis are <img decoding=\"async\" style=\"height: 27px; width: 48px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image155.png\" \/>.<\/p>\n<p><strong>(ii) <\/strong>If the tangent parallel to <img decoding=\"async\" style=\"height: 17px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image145.png\" \/>axis.<\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0Slope of the tangent = <img decoding=\"async\" style=\"height: 16px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image156.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image157.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image077.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0 From eq. (ii),\u00a0 <img decoding=\"async\" style=\"height: 41px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image158.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image159.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0 From eq. (i),\u00a0 <img decoding=\"async\" style=\"height: 44px; width: 44px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image160.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 44px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image161.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 19px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image162.png\" \/><\/p>\n<p>Therefore, the points on curve (i) where tangents are parallel to <img decoding=\"async\" style=\"height: 17px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image145.png\" \/>axis are <img decoding=\"async\" style=\"height: 27px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image163.png\" \/>.<\/p>\n<hr \/>\n<h3><span class=\"ez-toc-section\" id=\"14_Find_the_equation_of_the_tangents_and_normal_to_the_given_curves_at_the_indicated_points\"><\/span><strong>14. Find the equation of the tangents and normal to the given curves at the indicated points:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>(i) <img decoding=\"async\" style=\"height: 24px; width: 183px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image164.png\" \/>\u00a0at (0, 5)<\/strong><\/p>\n<p><strong>(ii) <img decoding=\"async\" style=\"height: 24px; width: 183px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image164.png\" \/>\u00a0at (1, 3)<\/strong><\/p>\n<p><strong>(iii) <img decoding=\"async\" style=\"height: 24px; width: 42px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image165.png\" \/>\u00a0at (1, 1)<\/strong><\/p>\n<p><strong>(iv) <img decoding=\"async\" style=\"height: 24px; width: 44px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image166.png\" \/>\u00a0at (0, 0)<\/strong><\/p>\n<p><strong>Ans. (i)<\/strong> Equation of the curve <img decoding=\"async\" style=\"height: 24px; width: 183px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image164.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 41px; width: 172px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image167.png\" \/><\/p>\n<p>Now value of <img decoding=\"async\" style=\"height: 41px; width: 23px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image023.png\" \/>\u00a0at (0, 5)<\/p>\n<p>At <img decoding=\"async\" style=\"height: 21px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image119.png\" \/><img decoding=\"async\" style=\"height: 30px; width: 247px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image168.png\" \/>\u00a0(say)<\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0Slope of the normal at (0, 5) is <img decoding=\"async\" style=\"height: 41px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image169.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0Equation of the tangent at (0, 5) is <img decoding=\"async\" style=\"height: 27px; width: 109px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image170.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image171.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image172.png\" \/><\/p>\n<p>And Equation of the normal at (0, 5) is <img decoding=\"async\" style=\"height: 41px; width: 112px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image173.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image174.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 104px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image175.png\" \/><\/p>\n<p><strong>(ii)<\/strong> Equation of the curve <img decoding=\"async\" style=\"height: 24px; width: 183px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image164.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 41px; width: 172px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image167.png\" \/><\/p>\n<p>Now value of <img decoding=\"async\" style=\"height: 41px; width: 23px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image023.png\" \/>\u00a0at (1, 3)<\/p>\n<p>At <img decoding=\"async\" style=\"height: 21px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image176.png\" \/><img decoding=\"async\" style=\"height: 30px; width: 332px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image177.png\" \/>\u00a0(say)<\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0Slope of the normal at (1, 3) is <img decoding=\"async\" style=\"height: 41px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image178.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0Equation of the tangent at (1, 3) is <img decoding=\"async\" style=\"height: 27px; width: 101px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image179.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 90px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image180.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 66px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image181.png\" \/><\/p>\n<p>And Equation of the normal at (1, 3) is <img decoding=\"async\" style=\"height: 41px; width: 111px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image182.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 97px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image183.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image184.png\" \/><\/p>\n<p><strong>(iii)<\/strong> Equation of the curve <img decoding=\"async\" style=\"height: 24px; width: 42px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image165.png\" \/>\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image185.png\" \/><\/p>\n<p>Now value of <img decoding=\"async\" style=\"height: 41px; width: 23px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image023.png\" \/>\u00a0at (1, 1)<\/p>\n<p>At <img decoding=\"async\" style=\"height: 21px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image176.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 30px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image186.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 15px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image187.png\" \/>\u00a0(say)<\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0Slope of the normal at (1, 1) is <img decoding=\"async\" style=\"height: 41px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image188.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0Equation of the tangent at (1, 1) is <img decoding=\"async\" style=\"height: 27px; width: 97px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image189.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image190.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image191.png\" \/><\/p>\n<p>And Equation of the normal at (1, 1) is <img decoding=\"async\" style=\"height: 41px; width: 109px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image192.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 95px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image193.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 90px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image194.png\" \/><\/p>\n<p><strong>(iv)<\/strong> Equation of the curve <img decoding=\"async\" style=\"height: 24px; width: 44px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image166.png\" \/>\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image195.png\" \/><\/p>\n<p>Now value of <img decoding=\"async\" style=\"height: 41px; width: 23px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image023.png\" \/>\u00a0at (0, 0)<\/p>\n<p>At <img decoding=\"async\" style=\"height: 21px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image119.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 19px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image196.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 15px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image187.png\" \/>\u00a0(say)<\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0Equation of the tangent at (0, 0) is <img decoding=\"async\" style=\"height: 27px; width: 104px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image197.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image159.png\" \/><\/p>\n<p>And normal at (0, 0) is <img decoding=\"async\" style=\"height: 17px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image145.png\" \/>axis.<\/p>\n<p><strong>(v)<\/strong> Equation of the curves are <img decoding=\"async\" style=\"height: 21px; width: 115px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image198.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image199.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 41px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image200.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 191px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image201.png\" \/><\/p>\n<p>Slope of the tangent at <img decoding=\"async\" style=\"height: 41px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image202.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 41px; width: 109px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image203.png\" \/>\u00a0(say)<\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0Slope of the normal at <img decoding=\"async\" style=\"height: 41px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image202.png\" \/>\u00a0is <img decoding=\"async\" style=\"height: 41px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image204.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0Point <img decoding=\"async\" style=\"height: 27px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image024.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 27px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image205.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 45px; width: 97px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image206.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 48px; width: 73px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image207.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0Equation of the tangent is <img decoding=\"async\" style=\"height: 48px; width: 145px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image208.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 44px; width: 112px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image209.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 25px; width: 73px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image210.png\" \/><\/p>\n<p>And Equation of the normal is <img decoding=\"async\" style=\"height: 48px; width: 135px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image211.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 44px; width: 112px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image212.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 17px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image213.png\" \/><\/p>\n<hr \/>\n<h3><span class=\"ez-toc-section\" id=\"15_Find_the_equation_of_the_tangent_line_to_curve_which_is\"><\/span><strong>15. Find the equation of the tangent line to curve <img decoding=\"async\" style=\"height: 24px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image214.png\" \/>\u00a0which is:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>(a) parallel to the line <img decoding=\"async\" style=\"height: 21px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image215.png\" \/>\u00a0<\/strong><\/p>\n<p><strong>(b) perpendicular to the line <img decoding=\"async\" style=\"height: 21px; width: 88px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image216.png\" \/>\u00a0<\/strong><\/p>\n<p><strong>Ans. <\/strong>Given: \u00a0Equation of the curve <img decoding=\"async\" style=\"height: 24px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image214.png\" \/>\u2026\u2026\u2026.(i)<\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0 Slope of tangent = <img decoding=\"async\" style=\"height: 41px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image217.png\" \/>\u00a0\u2026\u2026.(ii)<\/p>\n<p><strong>(a)<\/strong> Slope of the line <img decoding=\"async\" style=\"height: 21px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image215.png\" \/>\u00a0is <img decoding=\"async\" style=\"height: 41px; width: 84px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image218.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0Slope of tangent parallel to this line is also = 2<\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0From eq. (ii),\u00a0 <img decoding=\"async\" style=\"height: 19px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image219.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image041.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0From eq. (i),\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 107px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image220.png\" \/><\/p>\n<p>Therefore, point of contact is (2, 7).<\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0Equation of the tangent at (2, 7) is <img decoding=\"async\" style=\"height: 27px; width: 105px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image221.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image222.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 90px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image223.png\" \/><\/p>\n<p><strong>(b)<\/strong> Slope of the line <img decoding=\"async\" style=\"height: 21px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image224.png\" \/>\u00a0is <img decoding=\"async\" style=\"height: 44px; width: 115px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image225.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 15px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image187.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0Slope of the required tangent perpendicular to this line = <img decoding=\"async\" style=\"height: 41px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image188.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0From eq. (ii),\u00a0 <img decoding=\"async\" style=\"height: 41px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image226.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 19px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image227.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image228.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0From eq. (i),\u00a0 <img decoding=\"async\" style=\"height: 41px; width: 95px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image229.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 41px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image230.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 41px; width: 31px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image231.png\" \/><\/p>\n<p>Therefore, point of contact is <img decoding=\"async\" style=\"height: 45px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image232.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0Equation of the required tangent is <img decoding=\"async\" style=\"height: 45px; width: 139px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image233.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 120px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image234.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 111px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image235.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 157px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image236.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 107px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image237.png\" \/><\/p>\n<hr \/>\n<h3><span class=\"ez-toc-section\" id=\"16_Show_that_the_tangents_to_the_curve_at_the_points_where_and_are_parallel\"><\/span><strong>16. Show that the tangents to the curve <img decoding=\"async\" style=\"height: 24px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image238.png\" \/>\u00a0at the points where <img decoding=\"async\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image041.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 19px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image239.png\" \/>\u00a0are parallel.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Ans. <\/strong>Given: Equation of the curve <img decoding=\"async\" style=\"height: 24px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image238.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0 Slope of tangent at <img decoding=\"async\" style=\"height: 27px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image024.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 41px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image240.png\" \/><\/p>\n<p>At the point <img decoding=\"async\" style=\"height: 21px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image103.png\" \/>\u00a0Slope of the tangent = <img decoding=\"async\" style=\"height: 30px; width: 131px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image241.png\" \/><\/p>\n<p>At the point <img decoding=\"async\" style=\"height: 21px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image105.png\" \/>\u00a0Slope of the tangent = <img decoding=\"async\" style=\"height: 30px; width: 140px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image242.png\" \/><\/p>\n<p>Since, the slopes of the two tangents are equal.<\/p>\n<p>Therefore, tangents at <img decoding=\"async\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image041.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 19px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image239.png\" \/>\u00a0are parallel.<\/p>\n<hr \/>\n<h3><span class=\"ez-toc-section\" id=\"17_Find_the_points_on_the_curve_at_which_the_slope_of_the_tangent_is_equal_to_the_coordinate_of_the_point\"><\/span><strong>17. Find the points on the curve <img decoding=\"async\" style=\"height: 24px; width: 42px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image165.png\" \/>\u00a0at which the slope of the tangent is equal to the <img decoding=\"async\" style=\"height: 17px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image145.png\" \/>coordinate of the point.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Ans. <\/strong>Given: Equation of the curve <img decoding=\"async\" style=\"height: 24px; width: 42px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image165.png\" \/>\u2026\u2026\u2026(i)<\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0 Slope of tangent at <img decoding=\"async\" style=\"height: 27px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image024.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 41px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image185.png\" \/>\u2026\u2026\u2026.(ii)<\/p>\n<p>According to question, Slope of the tangent = <img decoding=\"async\" style=\"height: 17px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image145.png\" \/>coordinate of the point<\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image243.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image244.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image245.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 44px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image246.png\" \/>\u00a0or <img decoding=\"async\" style=\"height: 19px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image247.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image151.png\" \/>\u00a0or <img decoding=\"async\" style=\"height: 19px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image045.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0From eq. (i), at <img decoding=\"async\" style=\"height: 21px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image119.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image159.png\" \/>\u00a0The point is (0, 0).<\/p>\n<p>And From eq. (i), at <img decoding=\"async\" style=\"height: 21px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image082.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image248.png\" \/>The point is (3, 27).<\/p>\n<p>Therefore, the required points are (0, 0) and (3, 27).<\/p>\n<hr \/>\n<h3><span class=\"ez-toc-section\" id=\"18_For_the_curve_find_all_point_at_which_the_tangent_passes_through_the_origin\"><\/span><strong>18. For the curve <img decoding=\"async\" style=\"height: 24px; width: 92px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image249.png\" \/>\u00a0find all point at which the tangent passes through the origin.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Ans. <\/strong>Given: Equation of the curve <img decoding=\"async\" style=\"height: 24px; width: 88px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image250.png\" \/>\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0 Slope of the tangent at <img decoding=\"async\" style=\"height: 27px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image024.png\" \/>\u00a0passing through origin (0, 0)<\/p>\n<p>= <img decoding=\"async\" style=\"height: 41px; width: 111px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image251.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 41px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image252.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 41px; width: 104px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image253.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 24px; width: 101px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image254.png\" \/><\/p>\n<p>Substituting this value of <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image255.png\" \/>\u00a0in eq. (i), we get,<\/p>\n<p><img decoding=\"async\" style=\"height: 21px; width: 151px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image256.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image257.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 30px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image258.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 51px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image259.png\" \/>\u00a0or <img decoding=\"async\" style=\"height: 21px; width: 63px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image260.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image151.png\" \/>\u00a0or <img decoding=\"async\" style=\"height: 19px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image261.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0From eq. (i), at <img decoding=\"async\" style=\"height: 21px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image119.png\" \/><img decoding=\"async\" style=\"height: 21px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image159.png\" \/><\/p>\n<p>From eq. (i), at <img decoding=\"async\" style=\"height: 21px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image176.png\" \/><img decoding=\"async\" style=\"height: 21px; width: 84px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image262.png\" \/><\/p>\n<p>From eq. (i), at <img decoding=\"async\" style=\"height: 21px; width: 48px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image084.png\" \/><img decoding=\"async\" style=\"height: 21px; width: 101px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image263.png\" \/><\/p>\n<p>Therefore, the required points are (0, 0), (1, 2) and <img decoding=\"async\" style=\"height: 27px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image264.png\" \/><\/p>\n<hr \/>\n<h3><span class=\"ez-toc-section\" id=\"19_Find_the_points_on_the_curve_at_which_the_tangents_are_parallel_to_axis\"><\/span><strong>19. Find the points on the curve <img decoding=\"async\" style=\"height: 24px; width: 126px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image265.png\" \/>\u00a0at which the tangents are parallel to <img decoding=\"async\" style=\"height: 15px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image039.png\" \/>axis.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Ans. <\/strong>Equation of the curve <img decoding=\"async\" style=\"height: 24px; width: 126px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image265.png\" \/>\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 119px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image266.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 97px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image267.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 47px; width: 137px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image268.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image269.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 41px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image077.png\" \/>\u00a0[tangent is parallel to <img decoding=\"async\" style=\"height: 15px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image039.png\" \/>axis]\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 44px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image270.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 19px; width: 35px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image138.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0From eq. (i), <img decoding=\"async\" style=\"height: 24px; width: 107px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image271.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 24px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image272.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image273.png\" \/><\/p>\n<p>Therefore, the required points are (1, 2) and <img decoding=\"async\" style=\"height: 27px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image274.png\" \/><\/p>\n<hr \/>\n<h3><span class=\"ez-toc-section\" id=\"20_Find_the_equation_of_the_normal_at_the_point_for_the_curve\"><\/span><strong>20. Find the equation of the normal at the point <img decoding=\"async\" style=\"height: 30px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image275.png\" \/>\u00a0for the curve <img decoding=\"async\" style=\"height: 24px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image276.png\" \/>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Ans. <\/strong>Given: Equation of the curve <img decoding=\"async\" style=\"height: 24px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image277.png\" \/>\u2026\u2026\u2026.(i)<\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 101px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image278.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image279.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 47px; width: 66px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image280.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0Slope of the tangent at the point <img decoding=\"async\" style=\"height: 30px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image275.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 52px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image281.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 41px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image282.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0Slope of the normal at the point <img decoding=\"async\" style=\"height: 30px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image275.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 41px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image283.png\" \/><\/p>\n<p>Equation of the normal at <img decoding=\"async\" style=\"height: 30px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image275.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 41px; width: 155px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image284.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 24px; width: 168px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image285.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 24px; width: 183px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image286.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 30px; width: 188px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image287.png\" \/><\/p>\n<hr \/>\n<h3><span class=\"ez-toc-section\" id=\"21_Find_the_equations_of_the_normal_to_the_curve_which_are_parallel_to_the_line\"><\/span><strong>21. Find the equations of the normal to the curve <img decoding=\"async\" style=\"height: 24px; width: 97px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image288.png\" \/>\u00a0which are parallel to the line <img decoding=\"async\" style=\"height: 21px; width: 101px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image289.png\" \/>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Ans. <\/strong>Given: Equation of the curve <img decoding=\"async\" style=\"height: 24px; width: 97px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image288.png\" \/>\u00a0\u2026.(i)<\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0 Slope of the tangent at <img decoding=\"async\" style=\"height: 27px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image024.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 41px; width: 84px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image290.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0Slope of the normal to the curve at <img decoding=\"async\" style=\"height: 27px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image024.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 41px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image291.png\" \/>\u2026\u2026\u2026.(ii)<\/p>\n<p>But Slope of the normal (given) = <img decoding=\"async\" style=\"height: 41px; width: 23px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image292.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image291.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 41px; width: 23px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image292.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image293.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image294.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4943\" style=\"height: 21px; width: 44px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image101.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 4942\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4941\" style=\"height: 19px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image102.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 4939\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0From eq. (i), at <img decoding=\"async\" id=\"Picture 4938\" style=\"height: 21px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image103.png\" \/><img decoding=\"async\" id=\"Picture 4937\" style=\"height: 21px; width: 112px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image295.png\" \/><\/p>\n<p>at <img decoding=\"async\" id=\"Picture 4936\" style=\"height: 21px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image105.png\" \/><img decoding=\"async\" id=\"Picture 4935\" style=\"height: 21px; width: 124px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image296.png\" \/><\/p>\n<p>Therefore, the points of contact are (2, 18) and <img decoding=\"async\" id=\"Picture 4934\" style=\"height: 27px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image297.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 4933\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0Equation of the normal at (2, 18) is <img decoding=\"async\" id=\"Picture 4932\" style=\"height: 41px; width: 121px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image298.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 4931\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4930\" style=\"height: 21px; width: 123px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image299.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 4929\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4928\" style=\"height: 21px; width: 113px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image300.png\" \/><\/p>\n<p>And Equation of the normal at <img decoding=\"async\" id=\"Picture 31\" style=\"height: 27px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image301.png\" \/>\u00a0is <img decoding=\"async\" id=\"Picture 30\" style=\"height: 41px; width: 115px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image302.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 29\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 28\" style=\"height: 21px; width: 113px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image303.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 27\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 26\" style=\"height: 21px; width: 104px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image304.png\" \/><\/p>\n<hr \/>\n<h3><span class=\"ez-toc-section\" id=\"22_Find_the_equation_of_the_tangent_and_normal_to_the_parabola_at_the_point\"><\/span><strong>22. Find the equation of the tangent and normal to the parabola <img decoding=\"async\" style=\"height: 24px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image305.png\" \/>\u00a0at the point <img decoding=\"async\" style=\"height: 30px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image306.png\" \/>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Ans. <\/strong>Given: Equation of the parabola <img decoding=\"async\" id=\"Picture 25\" style=\"height: 24px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image305.png\" \/>\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p><img decoding=\"async\" id=\"Picture 24\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0 Slope of the tangent at <img decoding=\"async\" id=\"Picture 23\" style=\"height: 27px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image024.png\" \/><\/p>\n<p>= <img decoding=\"async\" id=\"Picture 22\" style=\"height: 41px; width: 116px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image307.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 21\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 20\" style=\"height: 41px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image308.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 19\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 18\" style=\"height: 44px; width: 95px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image309.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 17\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0Slope of the tangent at the point <img decoding=\"async\" id=\"Picture 16\" style=\"height: 29px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image310.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 15\" style=\"height: 41px; width: 53px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image311.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 14\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0Slope of the normal = <img decoding=\"async\" id=\"Picture 13\" style=\"height: 16px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image312.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 12\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0Equation of the tangent at the point <img decoding=\"async\" id=\"Picture 11\" style=\"height: 29px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image310.png\" \/><\/p>\n<p>= <img decoding=\"async\" id=\"Picture 10\" style=\"height: 41px; width: 131px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image313.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 9\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 8\" style=\"height: 24px; width: 116px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image314.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 7\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 6\" style=\"height: 24px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image315.png\" \/><\/p>\n<p>And Equation of the normal at the point <img decoding=\"async\" id=\"Picture 5\" style=\"height: 29px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image310.png\" \/><\/p>\n<p>= <img decoding=\"async\" id=\"Picture 4\" style=\"height: 29px; width: 137px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image316.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 2\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 1\" style=\"height: 24px; width: 111px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image317.png\" \/><\/p>\n<hr \/>\n<h3><span class=\"ez-toc-section\" id=\"23_Prove_that_the_curves_and_cut_at_right_angles_if\"><\/span><strong>23. Prove that the curves <img decoding=\"async\" style=\"height: 24px; width: 44px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image318.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 21px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image319.png\" \/>\u00a0cut at right angles if <img decoding=\"async\" style=\"height: 21px; width: 54px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image320.png\" \/>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Ans. <\/strong>Given: Equations of the curves are <img decoding=\"async\" id=\"Picture 4786\" style=\"height: 24px; width: 44px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image318.png\" \/>\u00a0\u2026..(i) and <img decoding=\"async\" id=\"Picture 4785\" style=\"height: 21px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image319.png\" \/>\u2026\u2026\u2026.(ii)<\/p>\n<p>Substituting the value of <img decoding=\"async\" id=\"Picture 4784\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image321.png\" \/>\u00a0in eq. (ii), we get\u00a0 <img decoding=\"async\" id=\"Picture 4783\" style=\"height: 24px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image322.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 4782\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0 <img decoding=\"async\" id=\"Picture 4781\" style=\"height: 29px; width: 51px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image323.png\" \/><\/p>\n<p>Putting the value of <img decoding=\"async\" id=\"Picture 4780\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image255.png\" \/>\u00a0in eq. (i), we get\u00a0 <img decoding=\"async\" id=\"Picture 4779\" style=\"height: 43px; width: 107px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image324.png\" \/><\/p>\n<p>Therefore, the point of intersection <img decoding=\"async\" id=\"Picture 4778\" style=\"height: 27px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image024.png\" \/>\u00a0is = <img decoding=\"async\" id=\"Picture 4777\" style=\"height: 40px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image325.png\" \/>\u2026\u2026\u2026.(iii)<\/p>\n<p>Differentiating eq. (i) w.r.t <img decoding=\"async\" id=\"Picture 4776\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image321.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4775\" style=\"height: 41px; width: 64px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image326.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 4774\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0 <img decoding=\"async\" id=\"Picture 4773\" style=\"height: 44px; width: 89px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image327.png\" \/>\u00a0\u2026\u2026\u2026.(iv)<\/p>\n<p>Differentiating eq. (ii) w.r.t <img decoding=\"async\" id=\"Picture 4772\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image321.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4771\" style=\"height: 41px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image328.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 4770\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4769\" style=\"height: 41px; width: 92px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image329.png\" \/>\u2026\u2026\u2026.(v)<\/p>\n<p>According to the question, <img decoding=\"async\" id=\"Picture 4768\" style=\"height: 24px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image330.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 4991\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4990\" style=\"height: 45px; width: 95px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image331.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 4989\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4988\" style=\"height: 41px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image332.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 4987\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4986\" style=\"height: 19px; width: 43px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image333.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 4985\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4984\" style=\"height: 27px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image334.png\" \/>\u00a0 [From eq. (iii)]\n<p><img decoding=\"async\" id=\"Picture 4983\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4982\" style=\"height: 21px; width: 48px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image335.png\" \/>\u00a0 [Cubing both sides]\n<hr \/>\n<h3><span class=\"ez-toc-section\" id=\"24_Find_the_equation_of_the_tangent_and_normal_to_the_hyperbola_at_the_point\"><\/span><strong>24. Find the equation of the tangent and normal to the hyperbola <img decoding=\"async\" style=\"height: 44px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image336.png\" \/>\u00a0at the point <img decoding=\"async\" style=\"height: 27px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image337.png\" \/>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Ans. <\/strong>Given: Equation of the hyperbola <img decoding=\"async\" id=\"Picture 4981\" style=\"height: 44px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image338.png\" \/>\u2026\u2026\u2026.(i)<\/p>\n<p><img decoding=\"async\" id=\"Picture 4980\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0 <img decoding=\"async\" id=\"Picture 4979\" style=\"height: 41px; width: 107px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image339.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 4978\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0 <img decoding=\"async\" id=\"Picture 4977\" style=\"height: 41px; width: 101px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image340.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 4976\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4975\" style=\"height: 47px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image341.png\" \/>\u00a0\u2026\u2026\u2026.(ii)<\/p>\n<p><img decoding=\"async\" id=\"Picture 4974\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0Slope of tangent at <img decoding=\"async\" id=\"Picture 4973\" style=\"height: 27px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image342.png\" \/>\u00a0is <img decoding=\"async\" id=\"Picture 4972\" style=\"height: 48px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image343.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 4971\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0Equation of the tangent at <img decoding=\"async\" id=\"Picture 4970\" style=\"height: 27px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image342.png\" \/>\u00a0is <img decoding=\"async\" id=\"Picture 4969\" style=\"height: 48px; width: 141px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image344.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 4968\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4967\" style=\"height: 44px; width: 155px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image345.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 4966\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4965\" style=\"height: 44px; width: 133px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image346.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 4964\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4963\" style=\"height: 44px; width: 133px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image347.png\" \/>\u00a0 \u2026\u2026\u2026.(iii)<\/p>\n<p>Since <img decoding=\"async\" id=\"Picture 4962\" style=\"height: 27px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image342.png\" \/>\u00a0lies on the hyperbola (i), therefore, <img decoding=\"async\" id=\"Picture 4960\" style=\"height: 44px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image348.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 4959\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0 From eq. (iii),\u00a0 <img decoding=\"async\" id=\"Picture 4958\" style=\"height: 41px; width: 88px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image349.png\" \/><\/p>\n<p>Now, Slope of normal at <img decoding=\"async\" id=\"Picture 4957\" style=\"height: 27px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image342.png\" \/>\u00a0= <img decoding=\"async\" id=\"Picture 4956\" style=\"height: 48px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image350.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 4955\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0Equation of the normal at <img decoding=\"async\" id=\"Picture 4954\" style=\"height: 27px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image342.png\" \/>\u00a0is <img decoding=\"async\" id=\"Picture 4953\" style=\"height: 48px; width: 149px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image351.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 4952\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4951\" style=\"height: 25px; width: 213px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image352.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 4950\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4948\" style=\"height: 27px; width: 191px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image353.png\" \/><\/p>\n<p>Dividing both sides by <img decoding=\"async\" id=\"Picture 4947\" style=\"height: 25px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image354.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 4946\" style=\"height: 48px; width: 124px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image355.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 4945\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4944\" style=\"height: 48px; width: 135px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image356.png\" \/><\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_63-8\"><\/span>NCERT Solutions class 12 Maths Exercise 6.3<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"25_Find_the_equation_of_the_tangent_to_the_curve_which_is_parallel_to_the_line\"><\/span><strong>25. Find the equation of the tangent to the curve <img decoding=\"async\" style=\"height: 25px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image357.png\" \/>\u00a0which is parallel to the line <img decoding=\"async\" style=\"height: 21px; width: 101px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image358.png\" \/>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Ans. <\/strong>Given: Equation of the curve <img decoding=\"async\" id=\"Picture 4898\" style=\"height: 25px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image357.png\" \/>\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p><img decoding=\"async\" id=\"Picture 4897\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0 Slope of the tangent at point <img decoding=\"async\" id=\"Picture 2239\" style=\"height: 27px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image024.png\" \/>\u00a0is <img decoding=\"async\" id=\"Picture 2238\" style=\"height: 41px; width: 116px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image359.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 2237\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/> <img decoding=\"async\" id=\"Picture 2236\" style=\"height: 41px; width: 187px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image360.png\" \/><\/p>\n<p>= <img decoding=\"async\" id=\"Picture 2235\" style=\"height: 44px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image361.png\" \/>\u2026\u2026\u2026.(ii)<\/p>\n<p>Again slope of the line <img decoding=\"async\" id=\"Picture 2234\" style=\"height: 21px; width: 99px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image362.png\" \/>\u00a0is <img decoding=\"async\" id=\"Picture 2233\" style=\"height: 41px; width: 84px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image363.png\" \/>\u2026\u2026\u2026.(iii)<\/p>\n<h4><span class=\"ez-toc-section\" id=\"According_to_the_question_Parallel_lines_have_same_slope\"><\/span>According to the question, <img decoding=\"async\" id=\"Picture 2232\" style=\"height: 44px; width: 89px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image364.png\" \/>\u00a0[Parallel lines have same slope]<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><img decoding=\"async\" id=\"Picture 2231\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 2230\" style=\"height: 24px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image365.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 2229\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 2228\" style=\"height: 27px; width: 95px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image366.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 2227\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 2226\" style=\"height: 19px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image367.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 2225\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 2224\" style=\"height: 19px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image368.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 2223\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 2222\" style=\"height: 19px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image369.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 2221\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 2220\" style=\"height: 41px; width: 48px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image370.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 2219\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0 From eq. (i),\u00a0 <img decoding=\"async\" id=\"Picture 2218\" style=\"height: 51px; width: 107px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image371.png\" \/><\/p>\n<p>= <img decoding=\"async\" id=\"Picture 2217\" style=\"height: 47px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image372.png\" \/><\/p>\n<p>= <img decoding=\"async\" id=\"Picture 2216\" style=\"height: 47px; width: 135px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image373.png\" \/><\/p>\n<p>Therefore, point of contact is <img decoding=\"async\" id=\"Picture 2176\" style=\"height: 45px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image374.png\" \/>.<\/p>\n<p><img decoding=\"async\" id=\"Picture 4799\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0 Equation of the required tangent is <img decoding=\"async\" id=\"Picture 4797\" style=\"height: 45px; width: 121px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image375.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 4796\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4795\" style=\"height: 41px; width: 104px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image376.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 4794\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4793\" style=\"height: 41px; width: 104px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image377.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 4792\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4791\" style=\"height: 41px; width: 107px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image378.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 4790\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4789\" style=\"height: 21px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image379.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 4788\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 4787\" style=\"height: 21px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image380.png\" \/><\/p>\n<hr \/>\n<p><strong>Choose the correct answer in Exercises 26 and 27.<\/strong><\/p>\n<h3><span class=\"ez-toc-section\" id=\"26_The_slope_of_the_normal_to_the_curve_at_is\"><\/span><strong>26. The slope of the normal to the curve <img decoding=\"async\" style=\"height: 24px; width: 107px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image381.png\" \/>\u00a0at <img decoding=\"async\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image151.png\" \/>\u00a0is:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>(A) 3 <\/strong><\/p>\n<p><strong>(B) <img decoding=\"async\" style=\"height: 41px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image382.png\" \/><\/strong><\/p>\n<p><strong>(C) <img decoding=\"async\" style=\"height: 19px; width: 21px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image383.png\" \/>\u00a0<\/strong><\/p>\n<p><strong>(D) <img decoding=\"async\" style=\"height: 41px; width: 23px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image384.png\" \/>\u00a0<\/strong><\/p>\n<p><strong>Ans. <\/strong>Given: Equation of the curve <img decoding=\"async\" id=\"Picture 4909\" style=\"height: 24px; width: 107px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image381.png\" \/>\u2026\u2026\u2026.(i)<\/p>\n<p><img decoding=\"async\" id=\"Picture 4907\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0 Slope of the tangent at point <img decoding=\"async\" id=\"Picture 4906\" style=\"height: 27px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image024.png\" \/>\u00a0is <img decoding=\"async\" id=\"Picture 4904\" style=\"height: 41px; width: 111px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image385.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 4903\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0Slope of the tangent at <img decoding=\"async\" id=\"Picture 4902\" style=\"height: 21px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image119.png\" \/><img decoding=\"async\" id=\"Picture 4901\" style=\"height: 27px; width: 140px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image386.png\" \/>\u00a0(say)<\/p>\n<p><img decoding=\"async\" id=\"Picture 4900\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0Slope of the normal = <img decoding=\"async\" id=\"Picture 4899\" style=\"height: 41px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image387.png\" \/><\/p>\n<p>Therefore, option (D) if correct.<\/p>\n<hr \/>\n<h3><span class=\"ez-toc-section\" id=\"27_The_line_is_a_tangent_to_the_curve_at_the_point\"><\/span><strong>27. The line <img decoding=\"async\" style=\"height: 21px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image388.png\" \/>\u00a0is a tangent to the curve <img decoding=\"async\" style=\"height: 24px; width: 54px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image389.png\" \/>\u00a0at the point:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>(A) (1, 2)\u00a0 <\/strong><\/p>\n<p><strong>(B) (2, 1)\u00a0 <\/strong><\/p>\n<p><strong>(C) <img decoding=\"async\" style=\"height: 27px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image390.png\" \/>\u00a0<\/strong><\/p>\n<p><strong>(D) <img decoding=\"async\" style=\"height: 27px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image391.png\" \/>\u00a0<\/strong><\/p>\n<p><strong>Ans.<\/strong> Given: \u00a0Equation of the curve <img decoding=\"async\" id=\"Picture 564\" style=\"height: 24px; width: 53px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image392.png\" \/>\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p><img decoding=\"async\" id=\"Picture 565\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>Slope of the tangent at point <img decoding=\"async\" id=\"Picture 566\" style=\"height: 27px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image024.png\" \/>\u00a0is <img decoding=\"async\" id=\"Picture 567\" style=\"height: 41px; width: 63px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image393.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 568\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 569\" style=\"height: 44px; width: 51px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image394.png\" \/>\u2026\u2026\u2026.(ii)<\/p>\n<p><img decoding=\"async\" id=\"Picture 570\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0Slope of the line <img decoding=\"async\" id=\"Picture 571\" style=\"height: 21px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image388.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 572\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 573\" style=\"height: 21px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image395.png\" \/>\u00a0is <img decoding=\"async\" id=\"Picture 574\" style=\"height: 41px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image099.png\" \/>\u2026\u2026\u2026.(iii)<\/p>\n<p>From eq. (ii) and (iii),\u00a0 <img decoding=\"async\" id=\"Picture 575\" style=\"height: 44px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image396.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 576\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/><img decoding=\"async\" id=\"Picture 577\" style=\"height: 21px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image397.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 578\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image025.png\" \/>\u00a0From eq. (i),\u00a0 <img decoding=\"async\" id=\"Picture 579\" style=\"height: 19px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image398.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 580\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image053.png\" \/>\u00a0<img decoding=\"async\" id=\"Picture 581\" style=\"height: 19px; width: 35px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex6.3\/image138.png\" \/><\/p>\n<p>Therefore, required point is (1, 2).<\/p>\n<p>Therefore, option (A) is correct.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_63-9\"><\/span>NCERT Solutions class 12 Maths Exercise 6.3<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>NCERT Solutions Class 12 Maths PDF (Download) Free from myCBSEguide app and myCBSEguide website. Ncert solution class 12 Maths includes text book solutions from both part 1 and part 2. NCERT Solutions for CBSE Class 12 Maths have total 20 chapters. 12 Maths NCERT Solutions in PDF for free Download on our website. Ncert Maths class 12 solutions PDF and Maths ncert class 12 PDF solutions with latest modifications and as per the latest CBSE syllabus are only available in myCBSEguide<\/p>\n<h2><span class=\"ez-toc-section\" id=\"CBSE_App_for_Class_12\"><\/span>CBSE App for Class 12<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>To download\u00a0NCERT Solutions for class 12 Physics, Chemistry, Biology, History, Political Science, Economics, Geography, Computer Science, Home Science, Accountancy, Business Studies and Home Science; do check myCBSEguide app or website. myCBSEguide provides sample papers with solution, test papers for chapter-wise practice, NCERT solutions, NCERT Exemplar solutions, quick revision notes for ready reference, CBSE guess papers and CBSE important question papers. Sample Paper all are made available through\u00a0<a href=\"https:\/\/play.google.com\/store\/apps\/details?id=in.techchefs.MyCBSEGuide&amp;referrer=utm_source%3Dmycbse_bottom%26utm_medium%3Dtext%26utm_campaign%3Dmycbseads\"><strong>the best app for CBSE students<\/strong><\/a>\u00a0and myCBSEguide website.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>NCERT Solutions class 12 Maths Exercise 6.3 Class 12 Maths Class book solutions are available in PDF format for free download. These ncert book chapter wise questions and answers are very helpful for CBSE board exam. CBSE recommends NCERT books and most of the questions in CBSE exam are asked from NCERT text books. Class &#8230; <a title=\"NCERT Solutions class 12 Maths Exercise 6.3\" class=\"read-more\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-6-3\/\" aria-label=\"More on NCERT Solutions class 12 Maths Exercise 6.3\">Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":-1,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1432,281],"tags":[283,1342,321,216],"class_list":["post-4546","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematics-cbse-class-12","category-ncert-solutions","tag-cbse-study-material","tag-class-12","tag-mathematics","tag-ncert-solutions"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.0 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>NCERT Solutions class 12 Maths Exercise 6.3 | myCBSEguide<\/title>\n<meta name=\"description\" content=\"NCERT Solutions class 12 Maths Exercise 6.3 in PDF format for free download. 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