{"id":4537,"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-11-3\/"},"modified":"2018-05-08T15:56:29","modified_gmt":"2018-05-08T10:26:29","slug":"ncert-solutions-class-12-maths-exercise-11-3","status":"publish","type":"post","link":"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-11-3\/","title":{"rendered":"NCERT Solutions class 12 Maths Exercise 11.3"},"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-11-3\/#NCERT_Solutions_class_12_Maths_Three_Dimensional_Geometry\" >NCERT Solutions class 12 Maths Three Dimensional Geometry\u00a0<\/a><ul class='ez-toc-list-level-4' ><li class='ez-toc-heading-level-4'><ul class='ez-toc-list-level-4' ><li class='ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-11-3\/#NCERT_Solutions_class_12_Maths_Exercise_113\" >NCERT Solutions class 12 Maths Exercise 11.3<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-11-3\/#2_Find_the_vector_equation_of_a_plane_which_is_at_a_distance_of_7_units_from_the_origin_and_normal_to_the_vector\" >2. Find the vector equation of a plane which is at a distance of 7 units from the origin and normal to the vector \u00a0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-11-3\/#NCERT_Solutions_class_12_Maths_Exercise_113-2\" >NCERT Solutions class 12 Maths Exercise 11.3<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-11-3\/#3_Find_the_Cartesian_equation_of_the_following_planes\" >3. Find the Cartesian equation of the following planes:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-11-3\/#NCERT_Solutions_class_12_Maths_Exercise_113-3\" >NCERT Solutions class 12 Maths Exercise 11.3<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-11-3\/#4_In_the_following_cases_find_the_coordinates_of_the_foot_of_the_perpendicular_drawn_from_the_origin\" >4. In the following cases, find the coordinates of the foot of the perpendicular drawn from the origin:<\/a><ul class='ez-toc-list-level-5' ><li class='ez-toc-heading-level-5'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-11-3\/#Hence_putting_in_equation_ii_the_coordinates_of_foot_of_the_perpendicular_is\" >Hence, putting \u00a0in equation (ii), the coordinates of foot of the perpendicular is<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-11-3\/#NCERT_Solutions_class_12_Maths_Exercise_113-4\" >NCERT Solutions class 12 Maths Exercise 11.3<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-11-3\/#5_Find_the_vector_and_Cartesian_equations_of_the_planes\" >5. Find the vector and Cartesian equations of the planes<\/a><ul class='ez-toc-list-level-5' ><li class='ez-toc-heading-level-5'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-11-3\/#_Direction_ratios_of_normal_to_the_plane_are_coefficients_of_in_are\" >\u00a0Direction ratios of normal to the plane are coefficients of \u00a0in \u00a0are<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-11-3\/#NCERT_Solutions_class_12_Maths_Exercise_113-5\" >NCERT Solutions class 12 Maths Exercise 11.3<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-11-3\/#6_Find_the_equations_of_the_planes_that_passes_through_three_points\" >6. Find the equations of the planes that passes through three points:<\/a><ul class='ez-toc-list-level-5' ><li class='ez-toc-heading-level-5'><a class=\"ez-toc-link ez-toc-heading-14\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-11-3\/#_Points_A_B_and_C_are_not_collinear_and_hence_the_unique_plane_can_be_drawn_through_the_three_given_collinear_points_ie\" >\u00a0 Points A, B and C are not collinear and hence the unique plane can be drawn through the three given collinear points, i.e.,<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-15\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-11-3\/#NCERT_Solutions_class_12_Maths_Exercise_113-6\" >NCERT Solutions class 12 Maths Exercise 11.3<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-16\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-11-3\/#7_Find_the_intercepts_cut_off_by_the_plane\" >7. Find the intercepts cut off by the plane \u00a0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-17\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-11-3\/#NCERT_Solutions_class_12_Maths_Exercise_113-7\" >NCERT Solutions class 12 Maths Exercise 11.3<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-18\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-11-3\/#8_Find_the_equation_of_the_plane_with_intercept_3_on_the_axis_and_parallel_to_ZOX_plane\" >8. Find the equation of the plane with intercept 3 on the \u00a0axis and parallel to ZOX plane.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-19\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-11-3\/#NCERT_Solutions_class_12_Maths_Exercise_113-8\" >NCERT Solutions class 12 Maths Exercise 11.3<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-20\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-11-3\/#9_Find_the_equation_of_the_plane_through_the_intersection_of_the_planes_and_and_the_point_2_2_1\" >9. Find the equation of the plane through the intersection of the planes \u00a0and\u00a0and the point (2, 2, 1).<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-21\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-11-3\/#NCERT_Solutions_class_12_Maths_Exercise_113-9\" >NCERT Solutions class 12 Maths Exercise 11.3<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-22\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-11-3\/#10_Find_the_vector_equation_of_the_plane_passing_through_the_intersection_of_the_planes_and_through_the_point_2_1_3\" >10. Find the vector equation of the plane passing through the intersection of the planes \u00a0\u00a0and through the point (2, 1, 3).<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-23\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-11-3\/#NCERT_Solutions_class_12_Maths_Exercise_113-10\" >NCERT Solutions class 12 Maths Exercise 11.3<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-24\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-11-3\/#11_Find_the_equation_of_the_plane_through_the_line_of_intersection_of_the_planes_and_which_is_perpendicular_to_the_plane\" >11. Find the equation of the plane through the line of intersection of the planes \u00a0and \u00a0which is perpendicular to the plane \u00a0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-25\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-11-3\/#NCERT_Solutions_class_12_Maths_Exercise_113-11\" >NCERT Solutions class 12 Maths Exercise 11.3<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-26\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-11-3\/#12_Find_the_angle_between_the_planes_whose_vector_equations_are_and\" >12. Find the angle between the planes whose vector equations are \u00a0and<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-27\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-11-3\/#NCERT_Solutions_class_12_Maths_Exercise_113-12\" >NCERT Solutions class 12 Maths Exercise 11.3<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-28\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-11-3\/#13_In_the_following_cases_determine_whether_the_given_planes_are_parallel_or_perpendicular_and_in_case_they_are_neither_find_the_angle_between_them\" >13. In the following cases, determine whether the given planes are parallel or perpendicular and in case they are neither, find the angle between them.<\/a><ul class='ez-toc-list-level-5' ><li class='ez-toc-heading-level-5'><a class=\"ez-toc-link ez-toc-heading-29\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-11-3\/#d_equations_of_the_given_planes_are_and\" >(d) equations of the given planes are \u00a0\u00a0and \u00a0<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 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-11-3\/#NCERT_Solutions_class_12_Maths_Exercise_113-13\" >NCERT Solutions class 12 Maths Exercise 11.3<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-31\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-11-3\/#14_In_the_following_cases_find_the_distances_of_each_of_the_given_points_from_the_corresponding_given_plane\" >14. In the following cases find the distances of each of the given points from the corresponding given plane:<\/a><ul class='ez-toc-list-level-5' ><li class='ez-toc-heading-level-5'><a class=\"ez-toc-link ez-toc-heading-32\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-11-3\/#Ans_a_Distance_of_course_perpendicular_of_the_point_0_0_0_from_the_plane\" >Ans. (a) Distance (of course perpendicular) of the point (0, 0, 0) from the plane \u00a0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-5'><a class=\"ez-toc-link ez-toc-heading-33\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-11-3\/#c_Length_of_perpendicular_from_the_point_on_the_plane_is\" >(c) Length of perpendicular from the point \u00a0on the plane \u00a0\u00a0\u00a0is<\/a><\/li><\/ul><\/li><\/ul><\/li><\/ul><\/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-11-3\/#NCERT_Solutions_class_12_Maths_Exercise_113-14\" >NCERT Solutions class 12 Maths Exercise 11.3<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-35\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-11-3\/#CBSE_App_for_Students\" >CBSE App for Students<\/a><\/li><\/ul><\/nav><\/div>\n<p>NCERT Solutions class 12 Maths Exercise 11.3 Class 12 Maths 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.<\/p>\n<p><strong>Download\u00a0<a href=\"https:\/\/mycbseguide.com\/downloads\/cbse-class-12-mathematics-relations-and-functions\/1285\/ncert-solutions\/5\/\">NCERT solutions for\u00a0Three Dimensional Geometry\u00a0<\/a>as 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%20Part-02%20.jpg\" alt=\"NCERT Solutions class 12 Maths Exercise 11.3\" width=\"200\" height=\"256\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Three_Dimensional_Geometry\"><\/span>NCERT Solutions class 12 Maths Three Dimensional Geometry<strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><strong>Formula for equation number 1 and 2<\/strong><\/p>\n<p><strong>If <img decoding=\"async\" style=\"height: 17px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image001.png\" \/>\u00a0is the length of perpendicular from the origin to a plane and <img decoding=\"async\" style=\"height: 24px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image002.png\" \/>\u00a0is a unit normal vector to the plane, then equation of the plane is <img decoding=\"async\" style=\"height: 27px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image003.png\" \/>\u00a0(where of course <img decoding=\"async\" style=\"height: 17px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image001.png\" \/>\u00a0being length is &gt; 0)<\/strong><\/p>\n<p><strong>1. In each of the following cases, determine the direction cosines of the normal to the plane and the distance from the origin.<\/strong><\/p>\n<p><strong>(a) <img decoding=\"async\" style=\"height: 17px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image004.png\" \/><\/strong><\/p>\n<p><strong>(b) <img decoding=\"async\" style=\"height: 21px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image005.png\" \/>\u00a0<\/strong><\/p>\n<p><strong>(c) <img decoding=\"async\" style=\"height: 21px; width: 97px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image006.png\" \/><\/strong><\/p>\n<p><strong>(d) <img decoding=\"async\" style=\"height: 21px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image007.png\" \/>\u00a0<\/strong><\/p>\n<p><strong>Ans. (a)<\/strong> Given: Equation of the plane is <img decoding=\"async\" style=\"height: 17px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image004.png\" \/><\/p>\n<p>In vector form it is <img decoding=\"async\" style=\"height: 23px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image008.png\" \/>where <img decoding=\"async\" style=\"height: 19px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image009.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 104px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image011.png\" \/>(here <img decoding=\"async\" style=\"height: 19px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image012.png\" \/>\u00a0= 2 &gt; 0)<\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 35px; width: 201px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image013.png\" \/><\/p>\n<p>Here, <img decoding=\"async\" style=\"height: 27px; width: 104px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image014.png\" \/>\u00a0= (Position vector of point P<img decoding=\"async\" style=\"height: 27px; width: 53px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image015.png\" \/>\u00a0)<\/p>\n<p>And <img decoding=\"async\" style=\"height: 27px; width: 95px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image016.png\" \/><\/p>\n<p>Now let us reduce <img decoding=\"async\" style=\"height: 23px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image008.png\" \/>\u00a0to <img decoding=\"async\" style=\"height: 27px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image017.png\" \/><\/p>\n<p>Now <img decoding=\"async\" style=\"height: 23px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 56px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image018.png\" \/>[Dividing both sides by <img decoding=\"async\" style=\"height: 32px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image019.png\" \/>\u00a0]\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 73px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image020.png\" \/>\u00a0where <img decoding=\"async\" style=\"height: 56px; width: 211px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image021.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 95px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image022.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 21px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image023.png\" \/><\/p>\n<p>Therefore, direction cosines of normal to the plane are coefficients of <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image024.png\" \/>\u00a0in <img decoding=\"async\" style=\"height: 24px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image002.png\" \/>, i.e., 0, 0, 1 and length of perpendicular from the origin to the plane is <img decoding=\"async\" style=\"height: 21px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image023.png\" \/>.<\/p>\n<p><strong>(b) <\/strong>Given: Equation of the plane is <img decoding=\"async\" style=\"height: 21px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image005.png\" \/>(here <img decoding=\"async\" style=\"height: 19px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image009.png\" \/>)<\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 35px; width: 183px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image025.png\" \/><\/p>\n<p>i.e., <img decoding=\"async\" style=\"height: 23px; width: 44px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image026.png\" \/>\u00a0where <img decoding=\"async\" style=\"height: 27px; width: 79px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image027.png\" \/><\/p>\n<p>Dividing both sides by <img decoding=\"async\" style=\"height: 32px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image019.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 35px; width: 153px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image028.png\" \/>, we get<\/p>\n<p><img decoding=\"async\" style=\"height: 56px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image029.png\" \/> <img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 44px; width: 88px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image030.png\" \/>\u00a0where <img decoding=\"async\" style=\"height: 56px; width: 188px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image031.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 44px; width: 149px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image032.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 44px; width: 53px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image033.png\" \/><\/p>\n<p>Therefore direction cosines of the normal to the plane are the coefficients of <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image024.png\" \/>\u00a0in <img decoding=\"async\" style=\"height: 24px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image002.png\" \/>, i.e., <img decoding=\"async\" style=\"height: 44px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image034.png\" \/>and length of perpendicular from the origin to the plane is <img decoding=\"async\" style=\"height: 44px; width: 53px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image033.png\" \/>.<\/p>\n<p><strong>(c)<\/strong> Equation of the plane is <img decoding=\"async\" style=\"height: 21px; width: 97px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image006.png\" \/>(here <img decoding=\"async\" style=\"height: 19px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image009.png\" \/>\u00a0)<\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 35px; width: 200px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image035.png\" \/><\/p>\n<p>i.e., <img decoding=\"async\" style=\"height: 23px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image036.png\" \/>\u00a0where <img decoding=\"async\" style=\"height: 27px; width: 95px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image037.png\" \/><\/p>\n<p>Dividing both sides by <img decoding=\"async\" style=\"height: 32px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image019.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 35px; width: 176px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image038.png\" \/>, we get<\/p>\n<p><img decoding=\"async\" style=\"height: 56px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image029.png\" \/> <img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 44px; width: 95px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image039.png\" \/>\u00a0where <img decoding=\"async\" style=\"height: 56px; width: 212px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image040.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 44px; width: 172px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image041.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 44px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image042.png\" \/><\/p>\n<p>Therefore direction cosines of the normal to the plane are the coefficients of <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image024.png\" \/>\u00a0in <img decoding=\"async\" style=\"height: 24px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image002.png\" \/>, i.e., <img decoding=\"async\" style=\"height: 44px; width: 107px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image043.png\" \/>and length of perpendicular from the origin to the plane is <img decoding=\"async\" style=\"height: 44px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image042.png\" \/>.<\/p>\n<p><strong>(d)<\/strong> Given: Equation of the plane is <img decoding=\"async\" style=\"height: 21px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image007.png\" \/><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 55px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image044.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 53px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image045.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 128px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image046.png\" \/>\u00a0 (here <img decoding=\"async\" style=\"height: 19px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image009.png\" \/>\u00a0)<\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 35px; width: 207px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image047.png\" \/><\/p>\n<p>i.e., <img decoding=\"async\" style=\"height: 23px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image048.png\" \/>\u00a0where <img decoding=\"async\" style=\"height: 27px; width: 95px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image049.png\" \/><\/p>\n<p>Dividing both sides by <img decoding=\"async\" style=\"height: 32px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image019.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 35px; width: 201px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image050.png\" \/>, we get<\/p>\n<p><img decoding=\"async\" style=\"height: 56px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image029.png\" \/> <img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image051.png\" \/>\u00a0where <img decoding=\"async\" style=\"height: 56px; width: 228px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image052.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 41px; width: 196px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image053.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 41px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image054.png\" \/><\/p>\n<p>Therefore direction cosines of the normal to the plane are the coefficients of <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image024.png\" \/>\u00a0in <img decoding=\"async\" style=\"height: 24px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image002.png\" \/>, i.e., <img decoding=\"async\" style=\"height: 21px; width: 48px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image055.png\" \/>and length of perpendicular from the origin to the plane is <img decoding=\"async\" style=\"height: 41px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image054.png\" \/>.<\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_113\"><\/span>NCERT Solutions class 12 Maths Exercise 11.3<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"2_Find_the_vector_equation_of_a_plane_which_is_at_a_distance_of_7_units_from_the_origin_and_normal_to_the_vector\"><\/span><strong>2. Find the vector equation of a plane which is at a distance of 7 units from the origin and normal to the vector <\/strong><img decoding=\"async\" style=\"height: 27px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image056.png\" \/><strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>Here <img decoding=\"async\" style=\"height: 27px; width: 101px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/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\/Ex11.3\/image058.png\" \/>\u00a0 The unit vector perpendicular to the plane is<\/p>\n<p><img decoding=\"async\" style=\"height: 57px; width: 283px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image059.png\" \/><\/p>\n<p>Also <img decoding=\"async\" style=\"height: 21px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image060.png\" \/>(given)<\/p>\n<p>Therefore the equation of the required plane is <img decoding=\"async\" style=\"height: 27px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image017.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 48px; width: 116px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image061.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 35px; width: 153px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image062.png\" \/><\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_113-2\"><\/span>NCERT Solutions class 12 Maths Exercise 11.3<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"3_Find_the_Cartesian_equation_of_the_following_planes\"><\/span><strong>3. Find the Cartesian equation of the following planes:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>(a) <\/strong><img decoding=\"async\" style=\"height: 35px; width: 103px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image063.png\" \/><strong>\u00a0<\/strong><\/p>\n<p><strong>(b) <\/strong><img decoding=\"async\" style=\"height: 35px; width: 123px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image064.png\" \/><strong>\u00a0<\/strong><\/p>\n<p><strong>(c) <\/strong><img decoding=\"async\" style=\"height: 35px; width: 248px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image065.png\" \/><strong>\u00a0<\/strong><\/p>\n<p><strong>Ans. (a)<\/strong> Vector equation of the plane is <img decoding=\"async\" style=\"height: 35px; width: 103px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image063.png\" \/>\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p>Putting <img decoding=\"async\" style=\"height: 27px; width: 104px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image014.png\" \/>\u00a0in eq. (i) as in 3-D, Cartesian equation of the plane is <img decoding=\"async\" style=\"height: 35px; width: 185px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/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\/Ex11.3\/image010.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 27px; width: 151px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image067.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image068.png\" \/><\/p>\n<p><strong>(b)<\/strong> Since, <img decoding=\"async\" style=\"height: 21px; width: 12px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image069.png\" \/>\u00a0is the position vector of any arbitrary point P<img decoding=\"async\" style=\"height: 27px; width: 53px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image015.png\" \/>\u00a0on the plane.<\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image058.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 35px; width: 205px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image070.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 103px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image071.png\" \/><\/p>\n<p>which is the required Cartesian equation.<\/p>\n<p><strong>(c)<\/strong> Vector equation of the plane is <img decoding=\"async\" style=\"height: 35px; width: 248px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image065.png\" \/><\/p>\n<p>Since, Since, <img decoding=\"async\" style=\"height: 21px; width: 12px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image069.png\" \/>\u00a0is the position vector of any arbitrary point P<img decoding=\"async\" style=\"height: 27px; width: 53px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image015.png\" \/>\u00a0on the plane.<\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image058.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 35px; width: 328px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image072.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 227px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image073.png\" \/>\u00a0which is the required Cartesian equation.<\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_113-3\"><\/span>NCERT Solutions class 12 Maths Exercise 11.3<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"4_In_the_following_cases_find_the_coordinates_of_the_foot_of_the_perpendicular_drawn_from_the_origin\"><\/span><strong>4. In the following cases, find the coordinates of the foot of the perpendicular drawn from the origin:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>(a) <img decoding=\"async\" style=\"height: 21px; width: 135px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image074.png\" \/><\/strong><\/p>\n<p><strong>(b) <img decoding=\"async\" style=\"height: 21px; width: 97px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image075.png\" \/>\u00a0<\/strong><\/p>\n<p><strong>(c) <img decoding=\"async\" style=\"height: 21px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image005.png\" \/><\/strong><\/p>\n<p><strong>(d) <img decoding=\"async\" style=\"height: 21px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image007.png\" \/>\u00a0<\/strong><\/p>\n<p><strong>Ans. (a)<\/strong> Given: Equation of the plane is <img decoding=\"async\" style=\"height: 21px; width: 135px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image074.png\" \/>\u2026\u2026.(i) and point is O (0, 0, 0)<\/p>\n<p>Let M be the foot of the perpendicular drawn from the origin (0, 0, 0) to the given plane.<\/p>\n<p>Since, direction ratios of perpendicular OM to plane are coefficients of <img decoding=\"async\" style=\"height: 17px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image076.png\" \/>\u00a0in <img decoding=\"async\" style=\"height: 21px; width: 135px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image074.png\" \/>, i.e., 2, 3, 4 = <img decoding=\"async\" style=\"height: 21px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image077.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\/Ex11.3\/image058.png\" \/>\u00a0 Equation of the perpendicular OM is <img decoding=\"async\" style=\"height: 41px; width: 161px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image078.png\" \/>\u00a0(say)<\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 41px; width: 96px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image079.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 127px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/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\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 143px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image081.png\" \/><\/p>\n<p>Therefore, point M on this line OM is M<img decoding=\"async\" style=\"height: 27px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image082.png\" \/> \u2026\u2026\u2026..(ii)<\/p>\n<p>But point M lies on plane (i)<\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image058.png\" \/>\u00a0Putting <img decoding=\"async\" style=\"height: 21px; width: 143px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image081.png\" \/>\u00a0in eq. (i), we have<\/p>\n<p><img decoding=\"async\" style=\"height: 27px; width: 196px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image083.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 19px; width: 124px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image084.png\" \/> <img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 19px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image085.png\" \/><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image086.png\" \/><\/p>\n<h5><span class=\"ez-toc-section\" id=\"Hence_putting_in_equation_ii_the_coordinates_of_foot_of_the_perpendicular_is\"><\/span>Hence, putting <img decoding=\"async\" style=\"height: 41px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image086.png\" \/>\u00a0in equation (ii), the coordinates of foot of the perpendicular is <img decoding=\"async\" style=\"height: 45px; width: 96px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image087.png\" \/><span class=\"ez-toc-section-end\"><\/span><\/h5>\n<p><strong>(b)<\/strong> Given: Equation of the plane is <img decoding=\"async\" style=\"height: 21px; width: 97px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image075.png\" \/>\u2026\u2026.(i) and point is O (0, 0, 0)<\/p>\n<p>Let M be the foot of the perpendicular drawn from the origin (0, 0, 0) to the given plane.<\/p>\n<p>Since, direction ratios of perpendicular OM to plane are coefficients of <img decoding=\"async\" style=\"height: 17px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image076.png\" \/>\u00a0in <img decoding=\"async\" style=\"height: 21px; width: 97px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image075.png\" \/>, i.e., 0, 3, 4 = <img decoding=\"async\" style=\"height: 21px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image077.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\/Ex11.3\/image058.png\" \/>\u00a0 Equation of the perpendicular OM is <img decoding=\"async\" style=\"height: 41px; width: 161px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image088.png\" \/>\u00a0(say)<\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 41px; width: 96px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image089.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 127px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image090.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 132px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image091.png\" \/><\/p>\n<p>Therefore, point M on this line OM is M<img decoding=\"async\" style=\"height: 27px; width: 72px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image092.png\" \/> \u2026\u2026\u2026..(ii)<\/p>\n<p>But point M lies on plane (i)<\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image058.png\" \/>\u00a0Putting <img decoding=\"async\" style=\"height: 21px; width: 132px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image091.png\" \/>\u00a0in eq. (i), we have<\/p>\n<p><img decoding=\"async\" style=\"height: 27px; width: 135px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image093.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 19px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image094.png\" \/> <img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 19px; width: 55px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image095.png\" \/><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image096.png\" \/><\/p>\n<p>Hence, putting <img decoding=\"async\" style=\"height: 41px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image096.png\" \/>\u00a0in equation (ii), the coordinates of foot of the perpendicular is <img decoding=\"async\" style=\"height: 45px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image097.png\" \/><\/p>\n<p><strong>(c) <\/strong>Given: Equation of the plane is <img decoding=\"async\" style=\"height: 21px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image005.png\" \/>\u2026\u2026.(i) and point is O (0, 0, 0)<\/p>\n<p>Let M be the foot of the perpendicular drawn from the origin (0, 0, 0) to the given plane.<\/p>\n<p>Since, direction ratios of perpendicular OM to plane are coefficients of <img decoding=\"async\" style=\"height: 17px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image076.png\" \/>\u00a0in <img decoding=\"async\" style=\"height: 21px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image005.png\" \/>, i.e., 1, 1, 1 = <img decoding=\"async\" style=\"height: 21px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image077.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\/Ex11.3\/image058.png\" \/>\u00a0 Equation of the perpendicular OM is <img decoding=\"async\" style=\"height: 41px; width: 161px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image098.png\" \/>\u00a0(say)<\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 41px; width: 96px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image099.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 127px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/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\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 119px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image101.png\" \/><\/p>\n<p>Therefore, point M on this line OM is M<img decoding=\"async\" style=\"height: 27px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image102.png\" \/> \u2026\u2026\u2026..(ii)<\/p>\n<p>But point M lies on plane (i)<\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image058.png\" \/>\u00a0Putting <img decoding=\"async\" style=\"height: 21px; width: 119px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image101.png\" \/>\u00a0in eq. (i), we have<\/p>\n<p><img decoding=\"async\" style=\"height: 19px; width: 84px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image103.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 19px; width: 43px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image104.png\" \/> <img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image105.png\" \/><\/p>\n<p>Hence, putting <img decoding=\"async\" style=\"height: 41px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image105.png\" \/>\u00a0in equation (ii), the coordinates of foot of the perpendicular is <img decoding=\"async\" style=\"height: 45px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image106.png\" \/><\/p>\n<p><strong>(d) <\/strong>Given: Equation of the plane is <img decoding=\"async\" style=\"height: 21px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image007.png\" \/>\u2026\u2026.(i) and point is O (0, 0, 0)<\/p>\n<p>Let M be the foot of the perpendicular drawn from the origin (0, 0, 0) to the given plane.<\/p>\n<p>Since, direction ratios of perpendicular OM to plane are coefficients of <img decoding=\"async\" style=\"height: 17px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image076.png\" \/>\u00a0in <img decoding=\"async\" style=\"height: 21px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image007.png\" \/>, i.e., 0, 5, 0 = <img decoding=\"async\" style=\"height: 21px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image077.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\/Ex11.3\/image058.png\" \/>\u00a0 Equation of the perpendicular OM is <img decoding=\"async\" style=\"height: 41px; width: 161px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image107.png\" \/>\u00a0(say)<\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 41px; width: 96px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image108.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 127px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/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\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 123px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image110.png\" \/><\/p>\n<p>Therefore, point M on this line OM is M<img decoding=\"async\" style=\"height: 27px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image111.png\" \/> \u2026\u2026\u2026..(ii)<\/p>\n<p>But point M lies on plane (i)<\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image058.png\" \/>\u00a0Putting <img decoding=\"async\" style=\"height: 21px; width: 123px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image110.png\" \/>\u00a0in eq. (i), we have<\/p>\n<p><img decoding=\"async\" style=\"height: 19px; width: 119px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image112.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 19px; width: 63px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image113.png\" \/> <img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image114.png\" \/><\/p>\n<p>Hence, putting <img decoding=\"async\" style=\"height: 41px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image114.png\" \/>\u00a0in equation (ii), the coordinates of foot of the perpendicular is <img decoding=\"async\" style=\"height: 45px; width: 156px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image115.png\" \/>.<\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_113-4\"><\/span>NCERT Solutions class 12 Maths Exercise 11.3<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"5_Find_the_vector_and_Cartesian_equations_of_the_planes\"><\/span><strong>5. Find the vector and Cartesian equations of the planes<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>(a) that passes through the point <\/strong><img decoding=\"async\" style=\"height: 27px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image116.png\" \/><strong>\u00a0and the normal to the plane is <\/strong><img decoding=\"async\" style=\"height: 27px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image117.png\" \/><strong>\u00a0<\/strong><\/p>\n<p><strong>(b) that passes through the point (1, 4, 6) and the normal vector to the plane is <\/strong><img decoding=\"async\" style=\"height: 27px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image118.png\" \/><strong>\u00a0<\/strong><\/p>\n<p><strong>Ans. (a) <\/strong>Vector form: The given point on the plane is <img decoding=\"async\" style=\"height: 27px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image116.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image058.png\" \/>\u00a0 The position vector of the given point is <img decoding=\"async\" style=\"height: 29px; width: 164px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image119.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 24px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image120.png\" \/><\/p>\n<p>Also Normal vector to the plane is <img decoding=\"async\" style=\"height: 27px; width: 79px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image121.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image058.png\" \/>\u00a0Vector equation of the required line is <img decoding=\"async\" style=\"height: 32px; width: 84px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image122.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 23px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image123.png\" \/><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/><img decoding=\"async\" style=\"height: 23px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image124.png\" \/><\/p>\n<p>Putting the values of <img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image125.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image126.png\" \/>,<\/p>\n<p><img decoding=\"async\" style=\"height: 35px; width: 209px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image127.png\" \/> <img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 35px; width: 233px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image128.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 35px; width: 121px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image129.png\" \/> <img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 35px; width: 101px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image130.png\" \/><\/p>\n<p>Cartesian form: The plane passes through the point <img decoding=\"async\" style=\"height: 27px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image116.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 27px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image131.png\" \/><\/p>\n<p>Normal vector to the plane is <img decoding=\"async\" style=\"height: 27px; width: 79px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image121.png\" \/><\/p>\n<h5><span class=\"ez-toc-section\" id=\"_Direction_ratios_of_normal_to_the_plane_are_coefficients_of_in_are\"><\/span><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image058.png\" \/>\u00a0Direction ratios of normal to the plane are coefficients of <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image024.png\" \/>\u00a0in <img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image126.png\" \/>\u00a0are <img decoding=\"async\" style=\"height: 21px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image132.png\" \/><span class=\"ez-toc-section-end\"><\/span><\/h5>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image058.png\" \/>\u00a0Cartesian form of equation of plane is <img decoding=\"async\" style=\"height: 27px; width: 228px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image133.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 196px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image134.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 123px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image135.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image136.png\" \/><\/p>\n<p><strong>(b)<\/strong> Vector form: The given point on the plane is <img decoding=\"async\" style=\"height: 27px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image137.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image058.png\" \/>\u00a0 The position vector of the given point is <img decoding=\"async\" style=\"height: 29px; width: 156px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image138.png\" \/><\/p>\n<p>Also Normal vector to the plane is <img decoding=\"async\" style=\"height: 27px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image139.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image058.png\" \/>\u00a0Vector equation of the required line is <img decoding=\"async\" style=\"height: 32px; width: 84px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image122.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 23px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image123.png\" \/><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/><img decoding=\"async\" style=\"height: 23px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image124.png\" \/><\/p>\n<p>Putting the values of <img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image125.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image126.png\" \/>,<\/p>\n<p><img decoding=\"async\" style=\"height: 35px; width: 257px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image140.png\" \/> <img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 35px; width: 225px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/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\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 35px; width: 143px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image142.png\" \/> <img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 35px; width: 111px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image143.png\" \/><\/p>\n<p>Cartesian form: The plane passes through the point <img decoding=\"async\" style=\"height: 27px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image137.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 27px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image131.png\" \/><\/p>\n<p>Normal vector to the plane is <img decoding=\"async\" style=\"height: 27px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image139.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image058.png\" \/>\u00a0Direction ratios of normal to the plane are coefficients of <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image024.png\" \/>\u00a0in <img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image126.png\" \/>\u00a0are <img decoding=\"async\" style=\"height: 21px; width: 43px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image144.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image058.png\" \/>\u00a0Cartesian form of equation of plane is <img decoding=\"async\" style=\"height: 27px; width: 228px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image133.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 205px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image145.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 152px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/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\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 99px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image147.png\" \/><\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_113-5\"><\/span>NCERT Solutions class 12 Maths Exercise 11.3<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"6_Find_the_equations_of_the_planes_that_passes_through_three_points\"><\/span><strong>6. Find the equations of the planes that passes through three points:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>(a) <\/strong><img decoding=\"async\" style=\"height: 27px; width: 189px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image148.png\" \/><strong>\u00a0<\/strong><\/p>\n<p><strong>(b) <\/strong><img decoding=\"async\" style=\"height: 27px; width: 167px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image149.png\" \/><strong>\u00a0<\/strong><\/p>\n<p><strong>Ans. <\/strong>We know that through three collinear points A, B, C i.e., through a straight line, we can pass an infinite number of planes.<\/p>\n<p><strong>(a)<\/strong> The three given points are A<img decoding=\"async\" style=\"height: 27px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image150.png\" \/> B<img decoding=\"async\" style=\"height: 27px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image151.png\" \/> and C<img decoding=\"async\" style=\"height: 27px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image152.png\" \/><\/p>\n<p>Now direction ratios of line AB are <img decoding=\"async\" style=\"height: 21px; width: 105px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image153.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 163px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image154.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 24px; width: 109px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image155.png\" \/><\/p>\n<p>Again direction ratios of line BC are <img decoding=\"async\" style=\"height: 27px; width: 144px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image156.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 24px; width: 132px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image157.png\" \/><\/p>\n<p>Now\u00a0 <img decoding=\"async\" style=\"height: 45px; width: 181px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image158.png\" \/><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 45px; width: 171px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image159.png\" \/><\/p>\n<p>Since, <img decoding=\"async\" style=\"height: 45px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image160.png\" \/><\/p>\n<p>Therefore, line AB and BC are parallel and B is their common point.<\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image058.png\" \/>\u00a0 Points A, B and C are collinear and hence an infinite number of planes can be drawn through the three given collinear points.<\/p>\n<p><strong>(b)<\/strong> The three given points are A<img decoding=\"async\" style=\"height: 27px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image161.png\" \/> B<img decoding=\"async\" style=\"height: 27px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image162.png\" \/> and C<img decoding=\"async\" style=\"height: 27px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image163.png\" \/><\/p>\n<p>Now direction ratios of line AB are <img decoding=\"async\" style=\"height: 21px; width: 95px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image164.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 163px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image154.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 24px; width: 96px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image165.png\" \/><\/p>\n<p>Again direction ratios of line BC are <img decoding=\"async\" style=\"height: 21px; width: 117px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image166.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 24px; width: 124px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image167.png\" \/><\/p>\n<p>Now\u00a0 <img decoding=\"async\" style=\"height: 45px; width: 164px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image168.png\" \/><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 45px; width: 171px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image159.png\" \/><\/p>\n<p>Since, <img decoding=\"async\" style=\"height: 45px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image169.png\" \/><\/p>\n<h5><span class=\"ez-toc-section\" id=\"_Points_A_B_and_C_are_not_collinear_and_hence_the_unique_plane_can_be_drawn_through_the_three_given_collinear_points_ie\"><\/span><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image058.png\" \/>\u00a0 Points A, B and C are not collinear and hence the unique plane can be drawn through the three given collinear points, i.e.,<span class=\"ez-toc-section-end\"><\/span><\/h5>\n<p><img decoding=\"async\" style=\"height: 75px; width: 220px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/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\/Ex11.3\/image010.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 75px; width: 205px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/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\/Ex11.3\/image010.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 75px; width: 184px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image172.png\" \/><\/p>\n<p>Expanding along first row,<\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 280px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/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\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 180px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/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\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 157px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image175.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 127px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image176.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 104px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image177.png\" \/><\/p>\n<p>Hence the equation of required plane is <img decoding=\"async\" style=\"height: 21px; width: 104px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image177.png\" \/>.<\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_113-6\"><\/span>NCERT Solutions class 12 Maths Exercise 11.3<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"7_Find_the_intercepts_cut_off_by_the_plane\"><\/span><strong>7. Find the intercepts cut off by the plane <\/strong><img decoding=\"async\" style=\"height: 21px; width: 93px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image178.png\" \/><strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Equation of the plane is <img decoding=\"async\" style=\"height: 21px; width: 89px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image179.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 41px; width: 95px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image180.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 65px; width: 105px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image181.png\" \/><\/p>\n<p style=\"text-align: justify;\">Comparing with intercept form <img decoding=\"async\" style=\"height: 41px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image182.png\" \/>, we have <img decoding=\"async\" style=\"height: 41px; width: 115px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image183.png\" \/>\u00a0which are intercepts cut off by the plane on <img decoding=\"async\" style=\"height: 15px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image184.png\" \/>axis, <img decoding=\"async\" style=\"height: 17px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image185.png\" \/>axis and <img decoding=\"async\" style=\"height: 13px; width: 24px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image186.png\" \/>axis respectively.<\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_113-7\"><\/span>NCERT Solutions class 12 Maths Exercise 11.3<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"8_Find_the_equation_of_the_plane_with_intercept_3_on_the_axis_and_parallel_to_ZOX_plane\"><\/span><strong>8. Find the equation of the plane with intercept 3 on the <\/strong><img decoding=\"async\" style=\"height: 17px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image185.png\" \/><strong>\u00a0axis and parallel to ZOX plane.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Since equation of ZOX plane is <img decoding=\"async\" style=\"height: 21px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image187.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image058.png\" \/>\u00a0 Equation of any plane parallel to ZOX plane is <img decoding=\"async\" style=\"height: 21px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image188.png\" \/>\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\">[<img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image189.png\" \/>\u00a0Equation of any plane parallel to the plane <img decoding=\"async\" style=\"height: 21px; width: 125px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image190.png\" \/>\u00a0is <img decoding=\"async\" style=\"height: 21px; width: 124px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image191.png\" \/>\u00a0i.e., change only the constant term]\n<p style=\"text-align: justify;\">Now, Plane (i) makes an intercept 3 on the <img decoding=\"async\" style=\"height: 17px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image185.png\" \/>axis (<img decoding=\"async\" style=\"height: 19px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image192.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 19px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image193.png\" \/>) i.e., plane (i) passes through (0, 3, 0).<\/p>\n<p style=\"text-align: justify;\">Putting <img decoding=\"async\" style=\"height: 21px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image194.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 19px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image193.png\" \/>\u00a0in eq. (i),\u00a0 <img decoding=\"async\" style=\"height: 19px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image195.png\" \/><\/p>\n<p style=\"text-align: justify;\">Putting <img decoding=\"async\" style=\"height: 19px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image196.png\" \/>\u00a0in eq. (i), equation of required plane is <img decoding=\"async\" style=\"height: 21px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image197.png\" \/><\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_113-8\"><\/span>NCERT Solutions class 12 Maths Exercise 11.3<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"9_Find_the_equation_of_the_plane_through_the_intersection_of_the_planes_and_and_the_point_2_2_1\"><\/span><strong>9. Find the equation of the plane through the intersection of the planes <\/strong><img decoding=\"async\" style=\"height: 21px; width: 120px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image198.png\" \/><strong>\u00a0and<\/strong><img decoding=\"async\" style=\"height: 21px; width: 104px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image199.png\" \/><strong>\u00a0and the point (2, 2, 1).<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Equations of given planes are <img decoding=\"async\" style=\"height: 21px; width: 120px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image198.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 21px; width: 104px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image199.png\" \/><\/p>\n<p style=\"text-align: justify;\">Since, equation of any plane through the intersection of these two planes is<\/p>\n<p style=\"text-align: justify;\">\u00a0 L.H.S. of plane I + <img decoding=\"async\" style=\"height: 19px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image200.png\" \/>\u00a0(L.H.S. of plane II) = 0<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 233px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image201.png\" \/>\u00a0 \u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\">Now, required plane (i) passes through the point (2, 2, 1).<\/p>\n<p style=\"text-align: justify;\">Putting <img decoding=\"async\" style=\"height: 21px; width: 111px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image202.png\" \/>\u00a0in eq. (i),<\/p>\n<p style=\"text-align: justify;\">\u00a0 <img decoding=\"async\" style=\"height: 27px; width: 249px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image203.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 212px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image204.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 19px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image205.png\" \/><\/p>\n<p style=\"text-align: justify;\">\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image206.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now putting <img decoding=\"async\" style=\"height: 41px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image206.png\" \/>\u00a0in eq. (i) of required plane is<\/p>\n<p style=\"text-align: justify;\">\u00a0 <img decoding=\"async\" style=\"height: 41px; width: 243px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image207.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 247px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image208.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 127px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image209.png\" \/><\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_113-9\"><\/span>NCERT Solutions class 12 Maths Exercise 11.3<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"10_Find_the_vector_equation_of_the_plane_passing_through_the_intersection_of_the_planes_and_through_the_point_2_1_3\"><\/span><strong>10. Find the vector equation of the plane passing through the intersection of the planes <\/strong><img decoding=\"async\" style=\"height: 35px; width: 131px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image210.png\" \/><strong>\u00a0<\/strong><img decoding=\"async\" style=\"height: 35px; width: 125px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image211.png\" \/><strong>\u00a0and through the point (2, 1, 3).<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Equation of first plane is <img decoding=\"async\" style=\"height: 35px; width: 127px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image212.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 35px; width: 148px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image213.png\" \/> \u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\">Again equation of the second plane is <img decoding=\"async\" style=\"height: 35px; width: 125px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image211.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 35px; width: 147px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image214.png\" \/> \u2026\u2026\u2026.(ii)<\/p>\n<p style=\"text-align: justify;\">Since, equation of any plane passing through the line of intersection of two planes is<\/p>\n<p style=\"text-align: justify;\">\u00a0 L.H.S. of plane I + <img decoding=\"async\" style=\"height: 19px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image200.png\" \/>\u00a0(L.H.S. of plane II) = 0<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 37px; width: 305px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image215.png\" \/>\u00a0 \u2026\u2026\u2026.(iii)<\/p>\n<p style=\"text-align: justify;\">Now, the plane (iii) passes through the point (2, 1, 3) = <img decoding=\"async\" style=\"height: 27px; width: 53px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image015.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image058.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 27px; width: 185px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image216.png\" \/><\/p>\n<p style=\"text-align: justify;\">Putting this value of <img decoding=\"async\" style=\"height: 21px; width: 12px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image069.png\" \/>\u00a0in eq. (iii),<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 37px; width: 448px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image217.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 213px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image218.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 19px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image219.png\" \/> <img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 48px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image220.png\" \/><\/p>\n<p style=\"text-align: justify;\">Putting <img decoding=\"async\" style=\"height: 41px; width: 48px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image220.png\" \/>\u00a0in eq. (iii) of required plane is<\/p>\n<p style=\"text-align: justify;\">\u00a0 <img decoding=\"async\" style=\"height: 41px; width: 312px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image221.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 37px; width: 355px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image222.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 35px; width: 292px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image223.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 35px; width: 155px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image224.png\" \/><\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_113-10\"><\/span>NCERT Solutions class 12 Maths Exercise 11.3<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"11_Find_the_equation_of_the_plane_through_the_line_of_intersection_of_the_planes_and_which_is_perpendicular_to_the_plane\"><\/span><strong>11. Find the equation of the plane through the line of intersection of the planes <\/strong><img decoding=\"async\" style=\"height: 21px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image005.png\" \/><strong>\u00a0and <\/strong><img decoding=\"async\" style=\"height: 21px; width: 105px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image225.png\" \/><strong>\u00a0which is perpendicular to the plane <\/strong><img decoding=\"async\" style=\"height: 21px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image226.png\" \/><strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Equations of the given planes are <img decoding=\"async\" style=\"height: 21px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image005.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 21px; width: 105px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image225.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 101px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image227.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 21px; width: 128px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image228.png\" \/><\/p>\n<p style=\"text-align: justify;\">Since, equation of any plane passing through the line of intersection of two planes is<\/p>\n<p style=\"text-align: justify;\">\u00a0 L.H.S. of plane I + <img decoding=\"async\" style=\"height: 19px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image200.png\" \/>\u00a0(L.H.S. of plane II) = 0<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 239px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image229.png\" \/> \u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 253px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image230.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 287px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image231.png\" \/><\/p>\n<p style=\"text-align: justify;\">According to the question, this plane is perpendicular to the plane <img decoding=\"async\" style=\"height: 21px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image232.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image058.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 24px; width: 131px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image233.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 192px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image234.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 19px; width: 168px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image235.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 19px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image236.png\" \/> <img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image237.png\" \/><\/p>\n<p style=\"text-align: justify;\">Putting <img decoding=\"async\" style=\"height: 41px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image237.png\" \/>\u00a0in eq. (i) of required plane is<\/p>\n<p style=\"text-align: justify;\">\u00a0 <img decoding=\"async\" style=\"height: 41px; width: 247px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image238.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 237px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image239.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 19px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image240.png\" \/><\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_113-11\"><\/span>NCERT Solutions class 12 Maths Exercise 11.3<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"12_Find_the_angle_between_the_planes_whose_vector_equations_are_and\"><\/span><strong>12. Find the angle between the planes whose vector equations are <\/strong><img decoding=\"async\" style=\"height: 35px; width: 125px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image241.png\" \/><strong>\u00a0and <\/strong><img decoding=\"async\" style=\"height: 35px; width: 127px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image242.png\" \/><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Equation of one plane is <img decoding=\"async\" style=\"height: 35px; width: 125px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image241.png\" \/>\u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\">Comparing this equation with <img decoding=\"async\" style=\"height: 27px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image243.png\" \/>, we have<\/p>\n<p style=\"text-align: justify;\">Normal vector to plane (i) is <img decoding=\"async\" style=\"height: 28px; width: 107px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image244.png\" \/><\/p>\n<p style=\"text-align: justify;\">Again, equation of second plane is <img decoding=\"async\" style=\"height: 35px; width: 124px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image245.png\" \/>\u00a0\u2026\u2026\u2026.(ii)<\/p>\n<p style=\"text-align: justify;\">Comparing this equation with <img decoding=\"async\" style=\"height: 27px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image243.png\" \/>, we have<\/p>\n<p style=\"text-align: justify;\">Normal vector to plane (i) is <img decoding=\"async\" style=\"height: 28px; width: 105px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image246.png\" \/><\/p>\n<p style=\"text-align: justify;\">Let <img decoding=\"async\" style=\"height: 19px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image247.png\" \/>\u00a0be the acute angle between plane (i) and (ii).<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image058.png\" \/>\u00a0 angle between normals <img decoding=\"async\" style=\"height: 27px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image248.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 27px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image249.png\" \/>\u00a0to planes (i) and (ii) is also <img decoding=\"async\" style=\"height: 19px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image250.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image058.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 61px; width: 256px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image251.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" style=\"height: 47px; width: 71px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image252.png\" \/><\/p>\n<p style=\"text-align: justify;\">\u00a0= <img decoding=\"async\" style=\"height: 47px; width: 95px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image253.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 44px; width: 101px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image254.png\" \/><\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_113-12\"><\/span>NCERT Solutions class 12 Maths Exercise 11.3<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"13_In_the_following_cases_determine_whether_the_given_planes_are_parallel_or_perpendicular_and_in_case_they_are_neither_find_the_angle_between_them\"><\/span><strong>13. In the following cases, determine whether the given planes are parallel or perpendicular and in case they are neither, find the angle between them.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\"><strong>(a) <\/strong><img decoding=\"async\" style=\"height: 21px; width: 136px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image255.png\" \/><strong>\u00a0and <\/strong><img decoding=\"async\" style=\"height: 21px; width: 125px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image256.png\" \/><strong>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(b) <\/strong><img decoding=\"async\" style=\"height: 21px; width: 120px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image257.png\" \/><strong>\u00a0and <\/strong><img decoding=\"async\" style=\"height: 21px; width: 89px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image258.png\" \/><strong>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(c) <\/strong><img decoding=\"async\" style=\"height: 21px; width: 128px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image259.png\" \/><strong>\u00a0and <\/strong><img decoding=\"async\" style=\"height: 21px; width: 124px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image260.png\" \/><strong>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(d) <\/strong><img decoding=\"async\" style=\"height: 21px; width: 117px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image261.png\" \/><strong>\u00a0and <\/strong><img decoding=\"async\" style=\"height: 19px; width: 117px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image262.png\" \/><strong>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(e) <\/strong><img decoding=\"async\" style=\"height: 21px; width: 119px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image263.png\" \/><strong>\u00a0and <\/strong><img decoding=\"async\" style=\"height: 21px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image264.png\" \/><strong>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. (a)<\/strong> Equations of the given planes are <img decoding=\"async\" style=\"height: 21px; width: 136px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image255.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 155px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image265.png\" \/>\u00a0and<\/p>\n<p style=\"text-align: justify;\">\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 125px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image256.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 161px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image266.png\" \/><\/p>\n<p style=\"text-align: justify;\">Here, <img decoding=\"async\" style=\"height: 45px; width: 172px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image267.png\" \/><\/p>\n<p style=\"text-align: justify;\">Since <img decoding=\"async\" style=\"height: 45px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image169.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, the given two planes are not parallel.<\/p>\n<p style=\"text-align: justify;\">Again <img decoding=\"async\" style=\"height: 24px; width: 105px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image268.png\" \/>\u00a0= 21 \u2013 5 \u2013 60 = 21 \u2013 65 = \u201344<\/p>\n<p style=\"text-align: justify;\">Since <img decoding=\"async\" style=\"height: 24px; width: 131px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image269.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, the given two planes are not perpendicular.<\/p>\n<p style=\"text-align: justify;\">Now let <img decoding=\"async\" style=\"height: 19px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image247.png\" \/>\u00a0be the angle between the two planes.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image058.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 52px; width: 228px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image270.png\" \/><\/p>\n<p style=\"text-align: justify;\">\u00a0= <img decoding=\"async\" style=\"height: 59px; width: 276px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image271.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" style=\"height: 47px; width: 173px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image272.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" style=\"height: 47px; width: 79px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image273.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 41px; width: 29px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image274.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 41px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image275.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 45px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image276.png\" \/><\/p>\n<p style=\"text-align: justify;\"><strong>(b)<\/strong> equations of the given planes are <img decoding=\"async\" style=\"height: 21px; width: 120px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image257.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 155px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image265.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 21px; width: 89px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image258.png\" \/>\u00a0i.e., <img decoding=\"async\" style=\"height: 21px; width: 120px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image277.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 161px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image266.png\" \/><\/p>\n<p style=\"text-align: justify;\">Here, <img decoding=\"async\" style=\"height: 45px; width: 156px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image278.png\" \/><\/p>\n<p style=\"text-align: justify;\">Since <img decoding=\"async\" style=\"height: 45px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image169.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, the given two planes are not parallel.<\/p>\n<p style=\"text-align: justify;\">Again <img decoding=\"async\" style=\"height: 24px; width: 105px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image268.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 27px; width: 125px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image279.png\" \/>\u00a0= 2 \u2013 2 + 0 = 0<\/p>\n<p style=\"text-align: justify;\">Since <img decoding=\"async\" style=\"height: 24px; width: 131px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image233.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, the given two planes are perpendicular.<\/p>\n<p style=\"text-align: justify;\"><strong>(c)<\/strong> equations of the given planes are <img decoding=\"async\" style=\"height: 21px; width: 128px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image259.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 155px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image265.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 21px; width: 124px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image260.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 161px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image266.png\" \/><\/p>\n<p style=\"text-align: justify;\">Here, <img decoding=\"async\" style=\"height: 45px; width: 209px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image280.png\" \/><\/p>\n<p style=\"text-align: justify;\">Since <img decoding=\"async\" style=\"height: 45px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image160.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, the given two planes are parallel.<\/p>\n<h5 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"d_equations_of_the_given_planes_are_and\"><\/span><strong>(d)<\/strong> equations of the given planes are <img decoding=\"async\" style=\"height: 21px; width: 117px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image261.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 155px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image265.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 21px; width: 119px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image281.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 161px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image266.png\" \/><span class=\"ez-toc-section-end\"><\/span><\/h5>\n<p style=\"text-align: justify;\">Here, <img decoding=\"async\" style=\"height: 45px; width: 224px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image282.png\" \/><\/p>\n<p style=\"text-align: justify;\">Since <img decoding=\"async\" style=\"height: 45px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image160.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, the given two planes are parallel.<\/p>\n<p style=\"text-align: justify;\"><strong>(e)<\/strong> equations of the given planes are <img decoding=\"async\" style=\"height: 21px; width: 119px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image263.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 155px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image265.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 21px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image283.png\" \/>\u00a0i.e., <img decoding=\"async\" style=\"height: 21px; width: 112px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image284.png\" \/><img decoding=\"async\" style=\"height: 27px; width: 161px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image266.png\" \/><\/p>\n<p style=\"text-align: justify;\">Here, <img decoding=\"async\" style=\"height: 45px; width: 144px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image285.png\" \/><\/p>\n<p style=\"text-align: justify;\">Since <img decoding=\"async\" style=\"height: 45px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image169.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, the given two planes are not parallel.<\/p>\n<p style=\"text-align: justify;\">Again <img decoding=\"async\" style=\"height: 24px; width: 105px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image268.png\" \/>\u00a0= 4 x 0 + 8 x 1 + 1 x 1 = 0 + 8 + 1 = 9<\/p>\n<p style=\"text-align: justify;\">Since <img decoding=\"async\" style=\"height: 24px; width: 131px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image269.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, the given two planes are not perpendicular.<\/p>\n<p style=\"text-align: justify;\">Now let <img decoding=\"async\" style=\"height: 19px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image247.png\" \/>\u00a0be the angle between the two planes.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image058.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 52px; width: 228px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image270.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" style=\"height: 59px; width: 244px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image286.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" style=\"height: 47px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image287.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 47px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image288.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 44px; width: 28px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image289.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 21px; width: 48px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image290.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 50px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image291.png\" \/><\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_113-13\"><\/span>NCERT Solutions class 12 Maths Exercise 11.3<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"14_In_the_following_cases_find_the_distances_of_each_of_the_given_points_from_the_corresponding_given_plane\"><\/span><strong>14. In the following cases find the distances of each of the given points from the corresponding given plane:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\"><strong>\u00a0(a) Point (0, 0, 0)<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Plane <\/strong><img decoding=\"async\" style=\"height: 21px; width: 112px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image292.png\" \/><strong>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(b) Point <\/strong><img decoding=\"async\" style=\"height: 27px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image293.png\" \/><\/p>\n<p style=\"text-align: justify;\"><strong>Plane <\/strong><img decoding=\"async\" style=\"height: 21px; width: 120px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image294.png\" \/><strong>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(c) Point <\/strong><img decoding=\"async\" style=\"height: 27px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image295.png\" \/><\/p>\n<p style=\"text-align: justify;\"><strong>Plane <\/strong><img decoding=\"async\" style=\"height: 21px; width: 99px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image296.png\" \/><strong>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(d) Point <\/strong><img decoding=\"async\" style=\"height: 27px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image297.png\" \/><\/p>\n<p style=\"text-align: justify;\"><strong>Plane <\/strong><img decoding=\"async\" style=\"height: 21px; width: 128px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image298.png\" \/><strong>\u00a0<\/strong><\/p>\n<h5 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"Ans_a_Distance_of_course_perpendicular_of_the_point_0_0_0_from_the_plane\"><\/span><strong>Ans. (a)<\/strong> Distance (of course perpendicular) of the point (0, 0, 0) from the plane <img decoding=\"async\" style=\"height: 21px; width: 104px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image299.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/><span class=\"ez-toc-section-end\"><\/span><\/h5>\n<p style=\"text-align: justify;\">\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 133px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image300.png\" \/>\u00a0is<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 48px; width: 123px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image301.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" style=\"height: 59px; width: 157px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image302.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" style=\"height: 47px; width: 92px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image303.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 44px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image304.png\" \/><\/p>\n<p style=\"text-align: justify;\"><strong>(b)<\/strong> Length of perpendicular from the point <img decoding=\"async\" style=\"height: 27px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image293.png\" \/>\u00a0on the plane <img decoding=\"async\" style=\"height: 21px; width: 120px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image305.png\" \/>\u00a0is<\/p>\n<p style=\"text-align: justify;\">\u00a0<img decoding=\"async\" style=\"height: 48px; width: 123px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image301.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" style=\"height: 59px; width: 148px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image306.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" style=\"height: 47px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image307.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 44px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image308.png\" \/><\/p>\n<h5 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"c_Length_of_perpendicular_from_the_point_on_the_plane_is\"><\/span><strong>(c)<\/strong> Length of perpendicular from the point <img decoding=\"async\" style=\"height: 27px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image295.png\" \/>\u00a0on the plane <img decoding=\"async\" style=\"height: 21px; width: 99px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image296.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 120px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image309.png\" \/>\u00a0is<span class=\"ez-toc-section-end\"><\/span><\/h5>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 48px; width: 123px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image301.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" style=\"height: 59px; width: 157px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image310.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" style=\"height: 47px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image311.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 44px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image312.png\" \/><\/p>\n<p style=\"text-align: justify;\"><strong>(d)<\/strong> Length of perpendicular from the point <img decoding=\"async\" style=\"height: 27px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image297.png\" \/>\u00a0on the plane <img decoding=\"async\" style=\"height: 21px; width: 128px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image298.png\" \/>\u00a0is<\/p>\n<p style=\"text-align: justify;\">\u00a0<img decoding=\"async\" style=\"height: 48px; width: 123px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image301.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" style=\"height: 59px; width: 160px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image313.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" style=\"height: 47px; width: 79px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image314.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 44px; width: 92px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.3\/image315.png\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_113-14\"><\/span>NCERT Solutions class 12 Maths Exercise 11.3<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 13 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_Students\"><\/span>CBSE App for Students<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 11.3 Class 12 Maths 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 &#8230; <a title=\"NCERT Solutions class 12 Maths Exercise 11.3\" class=\"read-more\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-11-3\/\" aria-label=\"More on NCERT Solutions class 12 Maths Exercise 11.3\">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-4537","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 11.3 | myCBSEguide<\/title>\n<meta name=\"description\" content=\"NCERT Solutions class 12 Maths Exercise 11.3 in PDF format for free download. 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