{"id":4538,"date":"2016-05-16T11:49:00","date_gmt":"2016-05-16T06:19:00","guid":{"rendered":"http:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-9\/"},"modified":"2018-05-08T16:15:30","modified_gmt":"2018-05-08T10:45:30","slug":"ncert-solutions-class-12-maths-miscellaneous-9","status":"publish","type":"post","link":"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-9\/","title":{"rendered":"NCERT Solutions class 12 Maths Miscellaneous"},"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-miscellaneous-9\/#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-miscellaneous-9\/#NCERT_Solutions_class_12_Maths_Miscellaneous\" >NCERT Solutions class 12 Maths Miscellaneous<\/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-miscellaneous-9\/#2If_and_are_the_direction_cosines_of_two_mutually_perpendicular_lines_show_that_the_direction_cosines_of_the_line_perpendicular_to_both_of_these_are\" >2.If \u00a0and \u00a0are the direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of these are \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-miscellaneous-9\/#NCERT_Solutions_class_12_Maths_Miscellaneous-2\" >NCERT Solutions class 12 Maths Miscellaneous<\/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-miscellaneous-9\/#3Find_the_angle_between_the_lines_whose_direction_ratios_are_and\" >3.Find the angle between the lines whose direction ratios are \u00a0and \u00a0<\/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-miscellaneous-9\/#NCERT_Solutions_class_12_Maths_Miscellaneous-3\" >NCERT Solutions class 12 Maths Miscellaneous<\/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-miscellaneous-9\/#4Find_the_equation_of_the_line_parallel_to_axis_and_passing_through_the_origin\" >4.Find the equation of the line parallel to axis and passing through the origin.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-9\/#NCERT_Solutions_class_12_Maths_Miscellaneous-4\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/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-miscellaneous-9\/#5If_the_coordinates_of_the_points_A_B_C_D_be_1_2_3_4_5_7_and_2_9_2_respectively_then_find_the_angle_between_the_lines_AB_and_CD\" >5.If the coordinates of the points A, B, C, D be (1, 2, 3), (4, 5, 7), \u00a0and (2, 9, 2) respectively, then find the angle between the lines AB and CD.<\/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-miscellaneous-9\/#NCERT_Solutions_class_12_Maths_Miscellaneous-5\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-9\/#6If_the_lines_and_are_perpendicular_find_the_value_of\" >6.If the lines \u00a0and \u00a0are perpendicular, find the value of \u00a0<\/a><\/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-miscellaneous-9\/#NCERT_Solutions_class_12_Maths_Miscellaneous-6\" >NCERT Solutions class 12 Maths Miscellaneous<\/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-miscellaneous-9\/#7Find_the_vector_equation_of_the_line_passing_through_1_2_3_and_perpendicular_to_the_plane\" >7.Find the vector equation of the line passing through (1, 2, 3) and 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-14\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-9\/#NCERT_Solutions_class_12_Maths_Miscellaneous-7\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/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-miscellaneous-9\/#8Find_the_equation_of_the_plane_passing_through_and_parallel_to_the_plane\" >8.Find the equation of the plane passing through \u00a0and parallel 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-16\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-9\/#NCERT_Solutions_class_12_Maths_Miscellaneous-8\" >NCERT Solutions class 12 Maths Miscellaneous<\/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-miscellaneous-9\/#9Find_the_shortest_distance_between_lines_and\" >9.Find the shortest distance between lines \u00a0and \u00a0<\/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-miscellaneous-9\/#NCERT_Solutions_class_12_Maths_Miscellaneous-9\" >NCERT Solutions class 12 Maths Miscellaneous<\/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-miscellaneous-9\/#10Find_the_coordinates_of_the_point_where_the_line_through_5_1_6_and_3_4_1_crosses_the_YZ-plane\" >10.Find the coordinates of the point where the line through (5, 1, 6) and (3, 4, 1) crosses the YZ-plane.<\/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-miscellaneous-9\/#NCERT_Solutions_class_12_Maths_Miscellaneous-10\" >NCERT Solutions class 12 Maths Miscellaneous<\/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-miscellaneous-9\/#11Find_the_coordinates_of_the_point_where_the_line_through_5_1_6_and_3_4_1_crosses_the_ZX-plane\" >11.Find the coordinates of the point where the line through (5, 1, 6) and (3, 4, 1) crosses the ZX-plane.<\/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-miscellaneous-9\/#NCERT_Solutions_class_12_Maths_Miscellaneous-11\" >NCERT Solutions class 12 Maths Miscellaneous<\/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-miscellaneous-9\/#12Find_the_coordinates_of_the_point_where_the_line_through_and_crosses_the_plane\" >12.Find the coordinates of the point where the line through \u00a0and \u00a0crosses the plane \u00a0<\/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-miscellaneous-9\/#NCERT_Solutions_class_12_Maths_Miscellaneous-12\" >NCERT Solutions class 12 Maths Miscellaneous<\/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-miscellaneous-9\/#13Find_the_equation_of_the_plane_passing_through_the_point_and_perpendicular_to_each_of_the_planes_and\" >13.Find the equation of the plane passing through the point \u00a0and perpendicular to each of the planes \u00a0and \u00a0<\/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-miscellaneous-9\/#NCERT_Solutions_class_12_Maths_Miscellaneous-13\" >NCERT Solutions class 12 Maths Miscellaneous<\/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-miscellaneous-9\/#14_If_the_points_and_be_equidistant_from_the_plane_then_find_the_value_of\" >14. If the points \u00a0and \u00a0be equidistant from the plane \u00a0then find the value of \u00a0<\/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-miscellaneous-9\/#NCERT_Solutions_class_12_Maths_Miscellaneous-14\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-29\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-9\/#15Find_the_equation_of_the_plane_passing_through_the_line_of_intersection_of_the_planes_and_and_parallel_to_axis\" >15.Find the equation of the plane passing through the line of intersection of the planes \u00a0and \u00a0and parallel to axis.<\/a><\/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-miscellaneous-9\/#NCERT_Solutions_class_12_Maths_Miscellaneous-15\" >NCERT Solutions class 12 Maths Miscellaneous<\/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-miscellaneous-9\/#16If_O_be_the_origin_and_the_coordinates_of_P_be_then_find_the_equation_of_the_plane_passing_through_P_and_perpendicular_to_OP\" >16.If O be the origin and the coordinates of P be \u00a0then find the equation of the plane passing through P and perpendicular to OP.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-32\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-9\/#NCERT_Solutions_class_12_Maths_Miscellaneous-16\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-33\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-9\/#17Find_the_equation_of_the_plane_which_contains_the_line_of_intersection_of_the_planes_and_which_is_perpendicular_to_the_plane\" >17.Find the equation of the plane which contains the line of intersection of the planes \u00a0\u00a0and which is perpendicular to the plane<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-34\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-9\/#NCERT_Solutions_class_12_Maths_Miscellaneous-17\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-35\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-9\/#18Find_the_distance_of_the_point_from_the_point_of_intersection_of_the_line_and_the_plane\" >18.Find the distance of the point \u00a0from the point of intersection of the line \u00a0and the plane<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-36\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-9\/#NCERT_Solutions_class_12_Maths_Miscellaneous-18\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-37\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-9\/#19Find_the_vector_equation_of_the_line_passing_through_and_parallel_to_the_plane_and\" >19.Find the vector equation of the line passing through \u00a0and parallel to the plane \u00a0and<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-38\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-9\/#NCERT_Solutions_class_12_Maths_Miscellaneous-19\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-39\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-9\/#20Find_the_vector_equation_of_the_line_passing_through_the_point_and_perpendicular_to_the_two_lines_and\" >20.Find the vector equation of the line passing through the point \u00a0and perpendicular to the two lines: \u00a0and \u00a0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-40\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-9\/#NCERT_Solutions_class_12_Maths_Miscellaneous-20\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-41\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-9\/#21Prove_that_if_a_plane_has_the_intercepts_and_is_at_a_distance_of_units_from_the_origin_then\" >21.Prove that if a plane has the intercepts \u00a0and is at a distance of \u00a0units from the origin, then \u00a0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-42\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-9\/#NCERT_Solutions_class_12_Maths_Miscellaneous-21\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-43\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-9\/#Choose_the_correct_answer_in_Exercise_Q_22_and_23\" >Choose the correct answer in Exercise Q. 22 and 23.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-44\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-9\/#NCERT_Solutions_class_12_Maths_Miscellaneous-22\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-45\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-9\/#23_The_planes_and_are\" >23. The planes: \u00a0and \u00a0are<\/a><\/li><\/ul><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-46\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-9\/#NCERT_Solutions_class_12_Maths_Miscellaneous-23\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-47\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-9\/#CBSE_App_for_Students\" >CBSE App for Students<\/a><\/li><\/ul><\/nav><\/div>\n<p>NCERT Solutions class 12 Maths Miscellaneous 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 Miscellaneous\" width=\"202\" height=\"259\" \/><\/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>1.Show that the line joining the origin to the point (2, 1, 1) is perpendicular to the line determined by the points <\/strong><img decoding=\"async\" style=\"height: 27px; width: 124px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image001.png\" \/><strong>\u00a0<\/strong><\/p>\n<p><strong>Ans. <\/strong>We know that direction ratios of the line joining the origin (0, 0, 0) to the point are<\/p>\n<p><img decoding=\"async\" style=\"height: 24px; width: 137px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image002.png\" \/>\u00a0= 2 \u2013 0, 1 \u2013 0, 1 \u2013 0 = 2, 1, 1 = <img decoding=\"async\" style=\"height: 24px; width: 51px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image003.png\" \/><\/p>\n<p>Similarly, direction ratios of the line joining the points <img decoding=\"async\" style=\"height: 27px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image004.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 27px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image005.png\" \/>\u00a0are<\/p>\n<p><img decoding=\"async\" style=\"height: 24px; width: 137px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image002.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 27px; width: 131px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image006.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 21px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image007.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 24px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image008.png\" \/><\/p>\n<p>For these two lines,<img decoding=\"async\" style=\"height: 24px; width: 105px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image009.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 27px; width: 124px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image010.png\" \/>\u00a0= 2 \u2013 2 + 0 = 0<\/p>\n<p>Therefore, the two given lines are perpendicular to each other.<\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"2If_and_are_the_direction_cosines_of_two_mutually_perpendicular_lines_show_that_the_direction_cosines_of_the_line_perpendicular_to_both_of_these_are\"><\/span><strong>2.If <\/strong><img decoding=\"async\" style=\"height: 24px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image011.png\" \/><strong>\u00a0and <\/strong><img decoding=\"async\" style=\"height: 24px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image012.png\" \/><strong>\u00a0are the direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of these are <\/strong><img decoding=\"async\" style=\"height: 24px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image013.png\" \/><strong>\u00a0<\/strong><img decoding=\"async\" style=\"height: 24px; width: 136px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image014.png\" \/><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong><img decoding=\"async\" style=\"height: 24px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image011.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 24px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image012.png\" \/>\u00a0are direction cosines of two mutually perpendicular of two given lines L<sub>1<\/sub> and L<sub>2<\/sub>. (say)<\/p>\n<p>Let <img decoding=\"async\" style=\"height: 28px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image015.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 28px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image016.png\" \/>\u00a0be the unit vectors along these lines L<sub>1<\/sub> and L<sub>2<\/sub>.<\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 28px; width: 121px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image018.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 28px; width: 128px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image019.png\" \/><\/p>\n<p>Let L be the line perpendicular to both the lines L<sub>1<\/sub> and L<sub>2<\/sub> and let <img decoding=\"async\" style=\"height: 24px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image020.png\" \/>\u00a0be a unit vector along line L perpendicular both lines L<sub>1<\/sub> and L<sub>2<\/sub>.<\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image017.png\" \/>\u00a0Cross-product of two vectors = <img decoding=\"async\" style=\"height: 35px; width: 156px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image021.png\" \/><\/p>\n[<img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image022.png\" \/>\u00a0L1 <img decoding=\"async\" style=\"height: 17px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image023.png\" \/>\u00a0L2 (given, <img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image017.png\" \/>\u00a0angle between them is <img decoding=\"async\" style=\"height: 21px; width: 24px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image024.png\" \/>]\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 28px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image026.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 83px; width: 196px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image027.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 29px; width: 316px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image028.png\" \/><\/p>\n<p>Since,<img decoding=\"async\" style=\"height: 24px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image020.png\" \/>\u00a0is a unit vector, therefore its components are its direction cosines.<\/p>\n<p>Thus, direction cosines of <img decoding=\"async\" style=\"height: 24px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image020.png\" \/>\u00a0are <img decoding=\"async\" style=\"height: 24px; width: 209px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image029.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/>direction cosines of line L are <img decoding=\"async\" style=\"height: 24px; width: 209px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image029.png\" \/><\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-2\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"3Find_the_angle_between_the_lines_whose_direction_ratios_are_and\"><\/span><strong>3.Find the angle between the lines whose direction ratios are <\/strong><img decoding=\"async\" style=\"height: 21px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image030.png\" \/><strong>\u00a0and <\/strong><img decoding=\"async\" style=\"height: 21px; width: 108px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image031.png\" \/><strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>Direction ratios of one line are <img decoding=\"async\" style=\"height: 21px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image030.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/>A vector along this line is <img decoding=\"async\" style=\"height: 28px; width: 107px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image032.png\" \/><\/p>\n<p>Direction ratios of second line are <img decoding=\"async\" style=\"height: 21px; width: 105px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image033.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/>A vector along second line is <img decoding=\"async\" style=\"height: 29px; width: 216px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image034.png\" \/><\/p>\n<p>Let <img decoding=\"async\" style=\"height: 19px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image035.png\" \/>\u00a0be the angle between the two lines, then<\/p>\n<p><img decoding=\"async\" style=\"height: 61px; width: 97px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image036.png\" \/>\u00a0=<img decoding=\"async\" style=\"height: 56px; width: 279px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image037.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 53px; width: 279px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image038.png\" \/>\u00a0= 0 = <img decoding=\"async\" style=\"height: 21px; width: 48px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image039.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 21px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image040.png\" \/><\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-3\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"4Find_the_equation_of_the_line_parallel_to_axis_and_passing_through_the_origin\"><\/span><strong>4.Find the equation of the line parallel to <\/strong><img decoding=\"async\" style=\"height: 15px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image041.png\" \/><strong>axis and passing through the origin.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>We know that a unit vector along <img decoding=\"async\" style=\"height: 15px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image041.png\" \/>axis is <img decoding=\"async\" style=\"height: 27px; width: 92px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image042.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image017.png\" \/>\u00a0Direction cosines of <img decoding=\"async\" style=\"height: 15px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image041.png\" \/>axis are coefficients of <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image043.png\" \/>\u00a0in the unit vector<\/p>\n<p>i.e., 1, 0, 0 = <img decoding=\"async\" style=\"height: 21px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image044.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image017.png\" \/>Equation of the required line passing through the origin (0, 0, 0) and parallel to <img decoding=\"async\" style=\"height: 15px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image041.png\" \/>axis is <img decoding=\"async\" style=\"height: 41px; width: 135px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image045.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image046.png\" \/><\/p>\n<p>Vector equation of the required line is <img decoding=\"async\" style=\"height: 23px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image047.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 24px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image048.png\" \/>\u00a0[<img decoding=\"async\" style=\"height: 23px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image049.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 24px; width: 33px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image050.png\" \/>\u00a0]\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 24px; width: 44px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image051.png\" \/><\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-4\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"5If_the_coordinates_of_the_points_A_B_C_D_be_1_2_3_4_5_7_and_2_9_2_respectively_then_find_the_angle_between_the_lines_AB_and_CD\"><\/span><strong>5.If the coordinates of the points A, B, C, D be (1, 2, 3), (4, 5, 7), <\/strong><img decoding=\"async\" style=\"height: 27px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image052.png\" \/><strong>\u00a0and (2, 9, 2) respectively, then find the angle between the lines AB and CD.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>Given: Points A<img decoding=\"async\" style=\"height: 27px; width: 55px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image053.png\" \/> B<img decoding=\"async\" style=\"height: 27px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image054.png\" \/> C<img decoding=\"async\" style=\"height: 27px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image052.png\" \/> and D<img decoding=\"async\" style=\"height: 27px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image055.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image017.png\" \/>\u00a0Direction ratios of line AB are <img decoding=\"async\" style=\"height: 24px; width: 137px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image002.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 24px; width: 213px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image056.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image017.png\" \/>A vector along the line AB is <img decoding=\"async\" style=\"height: 28px; width: 105px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image057.png\" \/><\/p>\n<p>Similarly, direction ratios of line CD are <img decoding=\"async\" style=\"height: 24px; width: 137px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image002.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 27px; width: 266px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image058.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image017.png\" \/>A vector along the line AB is <img decoding=\"async\" style=\"height: 28px; width: 107px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image059.png\" \/><\/p>\n<p>Let <img decoding=\"async\" style=\"height: 19px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image035.png\" \/>\u00a0be the angle between the two lines, then<\/p>\n<p><img decoding=\"async\" style=\"height: 61px; width: 97px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image036.png\" \/>\u00a0=<img decoding=\"async\" style=\"height: 49px; width: 165px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image060.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 47px; width: 84px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image061.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 44px; width: 84px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image062.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 41px; width: 44px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image063.png\" \/>\u00a0= 1<\/p>\n<p>= <img decoding=\"async\" style=\"height: 21px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image064.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 21px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image065.png\" \/><\/p>\n<p>Therefore, lines AB and CD are parallel.<\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-5\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"6If_the_lines_and_are_perpendicular_find_the_value_of\"><\/span><strong>6.If the lines <\/strong><img decoding=\"async\" style=\"height: 41px; width: 132px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image066.png\" \/><strong>\u00a0and <\/strong><img decoding=\"async\" style=\"height: 41px; width: 129px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image067.png\" \/><strong>\u00a0are perpendicular, find the value of <\/strong><img decoding=\"async\" style=\"height: 19px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image068.png\" \/><strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>Given: Equation of one line is <img decoding=\"async\" style=\"height: 41px; width: 132px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image066.png\" \/><\/p>\n<p>Direction ratios of this line are its denominators, i.e., <img decoding=\"async\" style=\"height: 21px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image069.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 24px; width: 51px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image003.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image017.png\" \/>A vector along this line is <img decoding=\"async\" style=\"height: 28px; width: 124px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image070.png\" \/><\/p>\n<p>Again, equation of second line is <img decoding=\"async\" style=\"height: 41px; width: 129px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image071.png\" \/><\/p>\n<p>Direction ratios of this line are its denominators, i.e., <img decoding=\"async\" style=\"height: 21px; width: 53px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image072.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 24px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image008.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image017.png\" \/>A vector along this line is <img decoding=\"async\" style=\"height: 28px; width: 107px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image073.png\" \/><\/p>\n<p>Since these given lines are perpendicular.<\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image017.png\" \/><img decoding=\"async\" style=\"height: 27px; width: 173px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image074.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 203px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image075.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 19px; width: 113px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image076.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 19px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image077.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 41px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image078.png\" \/><\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-6\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"7Find_the_vector_equation_of_the_line_passing_through_1_2_3_and_perpendicular_to_the_plane\"><\/span><strong>7.Find the vector equation of the line passing through (1, 2, 3) and perpendicular to the plane <\/strong><img decoding=\"async\" style=\"height: 35px; width: 143px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image079.png\" \/><strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>The required line passes through the point P (1, 2, 3).<\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image017.png\" \/>\u00a0Position vector <img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image080.png\" \/>\u00a0(say) of point P is (1, 2, 3)<\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 95px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image081.png\" \/><\/p>\n<p>Equation of the given plane is <img decoding=\"async\" style=\"height: 35px; width: 140px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image082.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 35px; width: 127px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image083.png\" \/><\/p>\n<p>Comparing with <img decoding=\"async\" style=\"height: 25px; width: 53px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image084.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 95px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image085.png\" \/><\/p>\n<p>Since, the required line is perpendicular to the given plane, therefore, vector <img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image086.png\" \/>\u00a0along the required line is <img decoding=\"async\" style=\"height: 27px; width: 119px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image087.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image022.png\" \/>\u00a0Equation of the required line is <img decoding=\"async\" style=\"height: 23px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image047.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 35px; width: 197px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image088.png\" \/><\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-7\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"8Find_the_equation_of_the_plane_passing_through_and_parallel_to_the_plane\"><\/span><strong>8.Find the equation of the plane passing through <\/strong><img decoding=\"async\" style=\"height: 27px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image089.png\" \/><strong>\u00a0and parallel to the plane <\/strong><img decoding=\"async\" style=\"height: 35px; width: 107px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image090.png\" \/><strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>Equation of any plane parallel to the plane <img decoding=\"async\" style=\"height: 35px; width: 103px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image091.png\" \/>\u00a0is <img decoding=\"async\" style=\"height: 35px; width: 104px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image092.png\" \/>\u2026..(i)<\/p>\n<p>Plane (i) passes through <img decoding=\"async\" style=\"height: 27px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image089.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image017.png\" \/>Putting <img decoding=\"async\" style=\"height: 29px; width: 165px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image093.png\" \/>\u00a0in eq. (i), we get<\/p>\n<p><img decoding=\"async\" style=\"height: 35px; width: 185px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image094.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 140px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image095.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 19px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image096.png\" \/><\/p>\n<p>Putting the value of <img decoding=\"async\" style=\"height: 19px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image097.png\" \/>\u00a0in eq. (i), to get the required plane is<\/p>\n<p><img decoding=\"async\" style=\"height: 35px; width: 147px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image098.png\" \/><\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-8\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"9Find_the_shortest_distance_between_lines_and\"><\/span><strong>9.Find the shortest distance between lines <\/strong><img decoding=\"async\" style=\"height: 35px; width: 207px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image099.png\" \/><strong>\u00a0and <\/strong><img decoding=\"async\" style=\"height: 35px; width: 180px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image100.png\" \/><strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>Given: Vector equation of one line is <img decoding=\"async\" style=\"height: 35px; width: 207px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image099.png\" \/><\/p>\n<p>Comparing with <img decoding=\"async\" style=\"height: 27px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image101.png\" \/>\u00a0we get<\/p>\n<p><img decoding=\"async\" style=\"height: 28px; width: 108px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image102.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 28px; width: 99px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image103.png\" \/><\/p>\n<p>Again given: Vector equation of another line is <img decoding=\"async\" style=\"height: 35px; width: 184px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image104.png\" \/><\/p>\n<p>Comparing with <img decoding=\"async\" style=\"height: 27px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image105.png\" \/>\u00a0we get<\/p>\n<p><img decoding=\"async\" style=\"height: 28px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image106.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 28px; width: 107px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image107.png\" \/><\/p>\n<p>We know that length of shortest distance between two (skew) lines is <img decoding=\"async\" style=\"height: 64px; width: 121px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image108.png\" \/>\u00a0..(i)<\/p>\n<p>Now<img decoding=\"async\" style=\"height: 35px; width: 207px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image109.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 27px; width: 137px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image110.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 27px; width: 95px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image111.png\" \/><\/p>\n<p>Again<img decoding=\"async\" style=\"height: 83px; width: 173px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image112.png\" \/><\/p>\n<p>Expanding along first row,<\/p>\n<p>= <img decoding=\"async\" style=\"height: 29px; width: 293px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image113.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 32px; width: 305px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image114.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 19px; width: 135px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image115.png\" \/><\/p>\n<p>And<img decoding=\"async\" style=\"height: 36px; width: 183px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image116.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 24px; width: 172px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image117.png\" \/><\/p>\n<p>Putting these values in eq. (i), length of shortest distance = <img decoding=\"async\" style=\"height: 44px; width: 109px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image118.png\" \/><\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-9\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"10Find_the_coordinates_of_the_point_where_the_line_through_5_1_6_and_3_4_1_crosses_the_YZ-plane\"><\/span><strong>10.Find the coordinates of the point where the line through (5, 1, 6) and (3, 4, 1) crosses the YZ-plane.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>Given: A line through the points A (5, 1, 6) and B (3, 4, 1)<\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image017.png\" \/>\u00a0Direction ratios of this line AB are <img decoding=\"async\" style=\"height: 24px; width: 137px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image002.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/>\u00a03 \u2013 5, 4 \u2013 1, 1 \u2013 6 <img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 21px; width: 107px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image119.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image017.png\" \/>Equation of the line AB is <img decoding=\"async\" style=\"height: 41px; width: 149px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image120.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 41px; width: 132px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image121.png\" \/>\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p>Now we have to find the coordinates of the point where this line AB crosses the YZ-plane<\/p>\n<p>i.e., <img decoding=\"async\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image122.png\" \/>\u00a0\u2026\u2026\u2026.(ii)<\/p>\n<p>Putting <img decoding=\"async\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image122.png\" \/>\u00a0in eq. (i), we get<\/p>\n<p><img decoding=\"async\" style=\"height: 41px; width: 119px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image123.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 41px; width: 63px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image124.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 41px; width: 64px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image125.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 21px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image126.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 19px; width: 89px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image127.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 21px; width: 53px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image128.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 19px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image129.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 41px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image130.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 41px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image131.png\" \/><\/p>\n<p>Thus, required point is P<img decoding=\"async\" style=\"height: 45px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image132.png\" \/><\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-10\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"11Find_the_coordinates_of_the_point_where_the_line_through_5_1_6_and_3_4_1_crosses_the_ZX-plane\"><\/span><strong>11.Find the coordinates of the point where the line through (5, 1, 6) and (3, 4, 1) crosses the ZX-plane.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>Given: A line through the points A (5, 1, 6) and B (3, 4, 1)<\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image017.png\" \/>\u00a0Direction ratios of this line AB are <img decoding=\"async\" style=\"height: 24px; width: 137px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image002.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/>3 \u2013 5, 4 \u2013 1, 1 \u2013 6 <img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 21px; width: 108px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image133.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image017.png\" \/>Equation of the line AB is <img decoding=\"async\" style=\"height: 41px; width: 149px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image120.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 41px; width: 132px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image121.png\" \/>\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p>Now we have to find the coordinates of the point where this line AB crosses the ZX-plane<\/p>\n<p>i.e., <img decoding=\"async\" style=\"height: 21px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image134.png\" \/>\u00a0\u2026\u2026\u2026.(ii)<\/p>\n<p>Putting <img decoding=\"async\" style=\"height: 21px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image134.png\" \/>\u00a0in eq. (i), we get<\/p>\n<p><img decoding=\"async\" style=\"height: 41px; width: 119px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image135.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 41px; width: 71px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image136.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 41px; width: 71px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image137.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 19px; width: 72px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image138.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 19px; width: 71px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image139.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 19px; width: 51px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image140.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 19px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/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\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 41px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image142.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 41px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image143.png\" \/><\/p>\n<p>Thus, required point is P<img decoding=\"async\" style=\"height: 45px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image144.png\" \/><\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-11\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"12Find_the_coordinates_of_the_point_where_the_line_through_and_crosses_the_plane\"><\/span><strong>12.Find the coordinates of the point where the line through <\/strong><img decoding=\"async\" style=\"height: 27px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image145.png\" \/><strong>\u00a0and <\/strong><img decoding=\"async\" style=\"height: 27px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image146.png\" \/><strong>\u00a0crosses the plane <\/strong><img decoding=\"async\" style=\"height: 21px; width: 93px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image147.png\" \/><strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>Direction ratios of the line joining the points A<img decoding=\"async\" style=\"height: 27px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image145.png\" \/>\u00a0and B<img decoding=\"async\" style=\"height: 27px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image146.png\" \/>\u00a0are<\/p>\n<p><img decoding=\"async\" style=\"height: 27px; width: 155px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image148.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 21px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image149.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image017.png\" \/>\u00a0Equation of the line AB are <img decoding=\"async\" style=\"height: 41px; width: 135px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image150.png\" \/>\u00a0\u2026\u2026\u2026(i)<\/p>\n<p>Equation of the plane is <img decoding=\"async\" style=\"height: 21px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image151.png\" \/>\u00a0\u2026\u2026\u2026(ii)<\/p>\n<p>Now to find the point where line (i) crosses plane (ii),<\/p>\n<p>From eq. (i)<img decoding=\"async\" style=\"height: 41px; width: 160px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image152.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\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 21px; width: 73px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image153.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image154.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 19px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image155.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 21px; width: 224px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image156.png\" \/>\u00a0\u2026\u2026\u2026.(iii)<\/p>\n<p>Putting the values of <img decoding=\"async\" style=\"height: 17px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image157.png\" \/>\u00a0in eq. (ii), we get<\/p>\n<p><img decoding=\"async\" style=\"height: 27px; width: 224px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image158.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 19px; width: 168px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image159.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 19px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image160.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 19px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image161.png\" \/><\/p>\n<p>Putting <img decoding=\"async\" style=\"height: 19px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image161.png\" \/>\u00a0in eq. (iii), point of intersection of line (i) and plane (ii) is<\/p>\n<p><img decoding=\"async\" style=\"height: 21px; width: 295px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image162.png\" \/><\/p>\n<p>Thus, required point of intersection is <img decoding=\"async\" style=\"height: 27px; width: 63px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image163.png\" \/><\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-12\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"13Find_the_equation_of_the_plane_passing_through_the_point_and_perpendicular_to_each_of_the_planes_and\"><\/span><strong>13.Find the equation of the plane passing through the point <\/strong><img decoding=\"async\" style=\"height: 27px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image164.png\" \/><strong>\u00a0and perpendicular to each of the planes <\/strong><img decoding=\"async\" style=\"height: 21px; width: 97px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image165.png\" \/><strong>\u00a0and <\/strong><img decoding=\"async\" style=\"height: 21px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image166.png\" \/><strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>Since equation of any plane through the point <img decoding=\"async\" style=\"height: 27px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image164.png\" \/>\u00a0is <img decoding=\"async\" style=\"height: 27px; width: 228px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image167.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 211px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image168.png\" \/>\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 183px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image169.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 21px; width: 171px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image170.png\" \/><\/p>\n<p>This required plane is perpendicular to the plane <img decoding=\"async\" style=\"height: 21px; width: 97px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image165.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 143px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image171.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image017.png\" \/>Product of coefficients <img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 27px; width: 144px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image172.png\" \/>\u00a0\u2026\u2026\u2026.(ii)<\/p>\n<p>Again the required plane is perpendicular to the plane <img decoding=\"async\" style=\"height: 21px; width: 97px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image173.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image017.png\" \/>Product of coefficients <img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 27px; width: 143px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image174.png\" \/>\u00a0\u2026\u2026\u2026.(iii)<\/p>\n<p>Solving eq. (ii) and (iii), we get<\/p>\n<p><img decoding=\"async\" style=\"height: 41px; width: 129px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/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\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 41px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image176.png\" \/><\/p>\n<p>Putting these values of <img decoding=\"async\" style=\"height: 21px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image030.png\" \/>\u00a0in eq. (i), we get<\/p>\n<p><img decoding=\"async\" style=\"height: 27px; width: 219px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image177.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 21px; width: 188px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image178.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 21px; width: 144px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image179.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 21px; width: 135px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image180.png\" \/><\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-13\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"14_If_the_points_and_be_equidistant_from_the_plane_then_find_the_value_of\"><\/span><strong>14. If the points <\/strong><img decoding=\"async\" style=\"height: 27px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image181.png\" \/><strong>\u00a0and <\/strong><img decoding=\"async\" style=\"height: 27px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image182.png\" \/><strong>\u00a0be equidistant from the plane <\/strong><img decoding=\"async\" style=\"height: 35px; width: 164px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image183.png\" \/><strong>\u00a0then find the value of <\/strong><img decoding=\"async\" style=\"height: 17px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image184.png\" \/><strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>Equation of the given plane is <img decoding=\"async\" style=\"height: 35px; width: 156px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image185.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 35px; width: 243px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image186.png\" \/><\/p>\n[<img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image022.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 12px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image187.png\" \/>\u00a0= Position vector of any point <img decoding=\"async\" style=\"height: 27px; width: 53px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image188.png\" \/>\u00a0on the plane <img decoding=\"async\" style=\"height: 27px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image189.png\" \/>]\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 21px; width: 140px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image190.png\" \/>\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p>Also, the point <img decoding=\"async\" style=\"height: 27px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image181.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 27px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image182.png\" \/>\u00a0are equidistant from plane (i)<\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/>(Perpendicular) distance of point <img decoding=\"async\" style=\"height: 27px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image181.png\" \/>\u00a0from plane (i)<\/p>\n<p>= Distance of point <img decoding=\"async\" style=\"height: 27px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image182.png\" \/>\u00a0from plane (i)<\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 49px; width: 343px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image191.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 44px; width: 205px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image192.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 27px; width: 103px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image193.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 21px; width: 92px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image194.png\" \/>\u00a0[<img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image022.png\" \/>\u00a0If <img decoding=\"async\" style=\"height: 27px; width: 84px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image195.png\" \/>\u00a0then<img decoding=\"async\" style=\"height: 17px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image196.png\" \/>]\n<p>Taking positive sign,<img decoding=\"async\" style=\"height: 21px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image197.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 21px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image198.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 21px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image199.png\" \/><\/p>\n<p>Taking negative sign, <img decoding=\"async\" style=\"height: 21px; width: 92px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image200.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 21px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image201.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 41px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image202.png\" \/><\/p>\n<p>Hence, the values of <img decoding=\"async\" style=\"height: 17px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image203.png\" \/>\u00a0are 1 or <img decoding=\"async\" style=\"height: 41px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image204.png\" \/><\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-14\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"15Find_the_equation_of_the_plane_passing_through_the_line_of_intersection_of_the_planes_and_and_parallel_to_axis\"><\/span><strong>15.Find the equation of the plane passing through the line of intersection of the planes <\/strong><img decoding=\"async\" style=\"height: 35px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image205.png\" \/><strong>\u00a0and <\/strong><img decoding=\"async\" style=\"height: 35px; width: 140px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image206.png\" \/><strong>\u00a0and parallel to <\/strong><img decoding=\"async\" style=\"height: 15px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image041.png\" \/><strong>axis.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>Equation of one plane is <img decoding=\"async\" style=\"height: 35px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image205.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 35px; width: 121px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image207.png\" \/>\u2026\u2026\u2026.(i)<\/p>\n<p>Equation of the second plane is <img decoding=\"async\" style=\"height: 35px; width: 140px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image206.png\" \/>\u2026\u2026\u2026.(ii)<\/p>\n<p>Since, equation of any plane passing through the line intersection of these two planes is<\/p>\n<p>L.H.S. of I + <img decoding=\"async\" style=\"height: 19px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image097.png\" \/>\u00a0(L.H.S. of II) = 0<\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 37px; width: 273px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image208.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 35px; width: 267px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image209.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 37px; width: 259px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image210.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 35px; width: 243px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image211.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 35px; width: 279px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image212.png\" \/>\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p>Comparing <img decoding=\"async\" style=\"height: 25px; width: 53px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image213.png\" \/>\u00a0we have<\/p>\n<p><img decoding=\"async\" style=\"height: 29px; width: 224px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image214.png\" \/><\/p>\n<p>Now required plane (i) is parallel to <img decoding=\"async\" style=\"height: 15px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image041.png\" \/>axis (<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/>\u00a0a vector <img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image086.png\" \/>\u00a0along <img decoding=\"async\" style=\"height: 15px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image041.png\" \/>axis is <img decoding=\"async\" style=\"height: 24px; width: 33px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image050.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 27px; width: 72px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image215.png\" \/>) <img decoding=\"async\" style=\"height: 23px; width: 48px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image216.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 27px; width: 259px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image217.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 19px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image218.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 19px; width: 55px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image219.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 41px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image220.png\" \/><\/p>\n<p>Putting <img decoding=\"async\" style=\"height: 41px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image220.png\" \/>\u00a0in eq. (i), the equation of required plane,<\/p>\n<p><img decoding=\"async\" style=\"height: 48px; width: 337px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image221.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 48px; width: 257px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image222.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 45px; width: 117px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image223.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 35px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image224.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 35px; width: 200px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image225.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 21px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image226.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 21px; width: 99px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image227.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 21px; width: 89px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image228.png\" \/><\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-15\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"16If_O_be_the_origin_and_the_coordinates_of_P_be_then_find_the_equation_of_the_plane_passing_through_P_and_perpendicular_to_OP\"><\/span><strong>16.If O be the origin and the coordinates of P be <\/strong><img decoding=\"async\" style=\"height: 27px; width: 64px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image229.png\" \/><strong>\u00a0then find the equation of the plane passing through P and perpendicular to OP.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>Given: Origin O (0, 0, 0) and point P<img decoding=\"async\" style=\"height: 27px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image230.png\" \/><\/p>\n<p>To find: Equation of the plane passing through P<img decoding=\"async\" style=\"height: 27px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image230.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 27px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image231.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image017.png\" \/>\u00a0Direction ratios of normal OP to the plane are <img decoding=\"async\" style=\"height: 21px; width: 111px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image232.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 109px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image233.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image017.png\" \/>Equation of the required plane is <img decoding=\"async\" style=\"height: 27px; width: 228px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image234.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 27px; width: 207px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image235.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 21px; width: 160px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image236.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 21px; width: 125px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image237.png\" \/><\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-16\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"17Find_the_equation_of_the_plane_which_contains_the_line_of_intersection_of_the_planes_and_which_is_perpendicular_to_the_plane\"><\/span><strong>17.Find the equation of the plane which contains the line of intersection of the planes <\/strong><img decoding=\"async\" style=\"height: 35px; width: 144px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image238.png\" \/><strong>\u00a0<\/strong><img decoding=\"async\" style=\"height: 35px; width: 132px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image239.png\" \/><strong>\u00a0and which is perpendicular to the plane <\/strong><img decoding=\"async\" style=\"height: 35px; width: 149px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image240.png\" \/><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>Equation of any plane passing through (or containing) the line of intersection of the planes <img decoding=\"async\" style=\"height: 35px; width: 140px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image241.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 35px; width: 132px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image242.png\" \/>\u00a0is L.H.S. of I + <img decoding=\"async\" style=\"height: 19px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image097.png\" \/>\u00a0(L.H.S. of II) = 0<\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 37px; width: 284px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image243.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 35px; width: 277px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image244.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 37px; width: 269px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image245.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 35px; width: 252px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image246.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 35px; width: 279px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image247.png\" \/>\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p>Comparing with <img decoding=\"async\" style=\"height: 27px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image248.png\" \/>\u00a0we have,<img decoding=\"async\" style=\"height: 29px; width: 225px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image249.png\" \/><\/p>\n<p>Now plane (i) is perpendicular to the given plane <img decoding=\"async\" style=\"height: 35px; width: 147px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image250.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 35px; width: 123px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image251.png\" \/><\/p>\n<p>Comparing with <img decoding=\"async\" style=\"height: 27px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image252.png\" \/>\u00a0we have,<img decoding=\"async\" style=\"height: 28px; width: 108px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image253.png\" \/><\/p>\n<p>For perpendicular planes<img decoding=\"async\" style=\"height: 27px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image254.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 27px; width: 243px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image255.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 19px; width: 188px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image256.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 19px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image257.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 19px; width: 53px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image258.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 41px; width: 48px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image259.png\" \/><\/p>\n<p>Putting <img decoding=\"async\" style=\"height: 41px; width: 48px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image259.png\" \/>\u00a0in eq. (i), equation of required plane is<\/p>\n<p><img decoding=\"async\" style=\"height: 48px; width: 328px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image260.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 48px; width: 303px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image261.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 35px; width: 156px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image262.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 35px; width: 231px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image263.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 21px; width: 136px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image264.png\" \/><\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-17\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"18Find_the_distance_of_the_point_from_the_point_of_intersection_of_the_line_and_the_plane\"><\/span><strong>18.Find the distance of the point <\/strong><img decoding=\"async\" style=\"height: 27px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image265.png\" \/><strong>\u00a0from the point of intersection of the line <\/strong><img decoding=\"async\" style=\"height: 35px; width: 203px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image266.png\" \/><strong>\u00a0and the plane <\/strong><img decoding=\"async\" style=\"height: 35px; width: 105px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image267.png\" \/><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>Given: A point P (say) <img decoding=\"async\" style=\"height: 27px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image265.png\" \/><\/p>\n<p>and equation of the line <img decoding=\"async\" style=\"height: 35px; width: 207px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image268.png\" \/>\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p>equation of the plane is <img decoding=\"async\" style=\"height: 35px; width: 101px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image269.png\" \/><\/p>\n<p>Putting the value of <img decoding=\"async\" style=\"height: 21px; width: 12px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image187.png\" \/>\u00a0from eq. (i) in eq. (ii),<\/p>\n<p><img decoding=\"async\" style=\"height: 37px; width: 300px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image270.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 35px; width: 353px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image271.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 27px; width: 167px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image272.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 19px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image273.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 19px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image274.png\" \/><\/p>\n<p>Putting <img decoding=\"async\" style=\"height: 19px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image274.png\" \/>\u00a0in eq. (i),<img decoding=\"async\" style=\"height: 35px; width: 204px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image275.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 27px; width: 95px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image276.png\" \/><\/p>\n<p>Therefore, Point of intersection is <img decoding=\"async\" style=\"height: 27px; width: 55px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image277.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 27px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image278.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image017.png\" \/>\u00a0Distance of the given point P<img decoding=\"async\" style=\"height: 27px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image265.png\" \/>\u00a0from the point of intersection is<\/p>\n<p><img decoding=\"async\" style=\"height: 35px; width: 204px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image279.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 24px; width: 89px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image280.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 24px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image281.png\" \/><\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-18\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"19Find_the_vector_equation_of_the_line_passing_through_and_parallel_to_the_plane_and\"><\/span><strong>19.Find the vector equation of the line passing through <\/strong><img decoding=\"async\" style=\"height: 27px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image282.png\" \/><strong>\u00a0and parallel to the plane <\/strong><img decoding=\"async\" style=\"height: 35px; width: 111px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image283.png\" \/><strong>\u00a0and <\/strong><img decoding=\"async\" style=\"height: 35px; width: 113px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image284.png\" \/><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>The required line passes through the point A (1, 2, 3) = <img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image080.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image017.png\" \/><img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image080.png\" \/>\u00a0= Position vector of point A = <img decoding=\"async\" style=\"height: 27px; width: 71px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image285.png\" \/><\/p>\n<p>Let <img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image086.png\" \/>\u00a0be any vector along the required line.<\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image017.png\" \/>Vector equation of required line is <img decoding=\"async\" style=\"height: 23px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image047.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 125px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image286.png\" \/>\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p>Since required line is parallel to the plane <img decoding=\"async\" style=\"height: 35px; width: 111px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image283.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image017.png\" \/><img decoding=\"async\" style=\"height: 27px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image287.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 27px; width: 53px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image288.png\" \/><\/p>\n<p>Comparing with <img decoding=\"async\" style=\"height: 27px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image248.png\" \/>\u00a0we have,<img decoding=\"async\" style=\"height: 28px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image289.png\" \/><\/p>\n<p>And Comparing with <img decoding=\"async\" style=\"height: 27px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image252.png\" \/>\u00a0we have,<img decoding=\"async\" style=\"height: 28px; width: 92px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image290.png\" \/><\/p>\n<p>Since <img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image086.png\" \/>\u00a0is perpendicular to both <img decoding=\"async\" style=\"height: 27px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image291.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 27px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image292.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image017.png\" \/><img decoding=\"async\" style=\"height: 27px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image293.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 83px; width: 115px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image294.png\" \/><\/p>\n<p>Expanding along first row,<\/p>\n<p><img decoding=\"async\" style=\"height: 29px; width: 212px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image295.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 27px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image296.png\" \/><\/p>\n<p>Putting this value of <img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image086.png\" \/>\u00a0in eq. (i), vector equation of required line,<\/p>\n<p><img decoding=\"async\" style=\"height: 35px; width: 225px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image297.png\" \/><\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-19\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"20Find_the_vector_equation_of_the_line_passing_through_the_point_and_perpendicular_to_the_two_lines_and\"><\/span><strong>20.Find the vector equation of the line passing through the point <\/strong><img decoding=\"async\" style=\"height: 27px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image298.png\" \/><strong>\u00a0and perpendicular to the two lines: <\/strong><img decoding=\"async\" style=\"height: 41px; width: 148px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image299.png\" \/><strong>\u00a0and <\/strong><img decoding=\"async\" style=\"height: 41px; width: 153px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image300.png\" \/><strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>Given: A point on the required line is A<img decoding=\"async\" style=\"height: 27px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image298.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image017.png\" \/>\u00a0Position vector of point A is <img decoding=\"async\" style=\"height: 29px; width: 165px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image301.png\" \/><\/p>\n<p>Also given equations of two lines<\/p>\n<p><img decoding=\"async\" style=\"height: 41px; width: 148px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image299.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 41px; width: 149px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image302.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image017.png\" \/>Direction ratios of given two lines are <img decoding=\"async\" style=\"height: 28px; width: 112px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image303.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 28px; width: 105px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image304.png\" \/><\/p>\n<p>Now<img decoding=\"async\" style=\"height: 27px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image305.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 83px; width: 124px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image306.png\" \/><\/p>\n<p>Expanding along first row,<\/p>\n<p>= <img decoding=\"async\" style=\"height: 29px; width: 241px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image307.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 27px; width: 103px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image308.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 35px; width: 131px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image309.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image017.png\" \/>Equation of the required line is <img decoding=\"async\" style=\"height: 23px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image047.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 35px; width: 247px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image310.png\" \/><\/p>\n<p>Again replacing <img decoding=\"async\" style=\"height: 19px; width: 29px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image311.png\" \/>\u00a0by <img decoding=\"async\" style=\"height: 21px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image312.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 35px; width: 217px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image313.png\" \/><\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-20\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"21Prove_that_if_a_plane_has_the_intercepts_and_is_at_a_distance_of_units_from_the_origin_then\"><\/span><strong>21.Prove that if a plane has the intercepts <\/strong><img decoding=\"async\" style=\"height: 21px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image030.png\" \/><strong>\u00a0and is at a distance of <\/strong><img decoding=\"async\" style=\"height: 17px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image203.png\" \/><strong>\u00a0units from the origin, then <\/strong><img decoding=\"async\" style=\"height: 44px; width: 125px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image314.png\" \/><strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>We know that equation of plane making intercepts <img decoding=\"async\" style=\"height: 21px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image030.png\" \/>\u00a0(on the axes) is <img decoding=\"async\" style=\"height: 41px; width: 109px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image315.png\" \/><\/p>\n<p>Given: Perpendicular distance of the origin (0, 0, 0) from plane = <img decoding=\"async\" style=\"height: 17px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image203.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 48px; width: 123px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image316.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 88px; width: 125px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image317.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 68px; width: 125px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image318.png\" \/><\/p>\n<p>Squaring both sides,<img decoding=\"async\" style=\"height: 61px; width: 119px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image319.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 45px; width: 141px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image320.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 45px; width: 136px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image321.png\" \/><\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-21\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"Choose_the_correct_answer_in_Exercise_Q_22_and_23\"><\/span><strong>Choose the correct answer in Exercise Q. 22 and 23.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>22.Distance between the two planes: <\/strong><img decoding=\"async\" style=\"height: 21px; width: 107px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image322.png\" \/><strong>\u00a0and <\/strong><img decoding=\"async\" style=\"height: 21px; width: 112px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image323.png\" \/><strong>\u00a0is<\/strong><\/p>\n<p><strong>(A) 2 units(B) 4 units(C) 8 units(D) <img decoding=\"async\" style=\"height: 44px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image324.png\" \/>\u00a0units<\/strong><\/p>\n<p><strong>Ans. <\/strong>Equation of one plane is <img decoding=\"async\" style=\"height: 21px; width: 107px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image322.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 21px; width: 128px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image325.png\" \/><\/p>\n<p>Equation of second plane is <img decoding=\"async\" style=\"height: 21px; width: 112px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image323.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 21px; width: 135px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image326.png\" \/><\/p>\n<p>Here<img decoding=\"async\" style=\"height: 45px; width: 227px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image327.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\/11%20Misc\/image328.png\" \/>\u00a0therefore, the given two lines are parallel.<\/p>\n<p>We know that the distance of the parallel lines = <img decoding=\"async\" style=\"height: 48px; width: 92px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image329.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image025.png\" \/><img decoding=\"async\" style=\"height: 59px; width: 128px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image330.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 47px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image331.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 44px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image324.png\" \/><\/p>\n<p>Therefore, option (D) is correct.<\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-22\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"23_The_planes_and_are\"><\/span><strong>23. The planes: <\/strong><img decoding=\"async\" style=\"height: 21px; width: 99px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image332.png\" \/><strong>\u00a0and <\/strong><img decoding=\"async\" style=\"height: 21px; width: 124px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image333.png\" \/><strong>\u00a0are <\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>(A) Perpendicular(B) Parallel<\/strong><\/p>\n<p><strong>(C) intersect <img decoding=\"async\" style=\"height: 17px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image334.png\" \/>axis(D) passes through <img decoding=\"async\" style=\"height: 45px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image335.png\" \/>\u00a0<\/strong><\/p>\n<p><strong>Ans. <\/strong>Equations of the given planes are <img decoding=\"async\" style=\"height: 21px; width: 99px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image332.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 151px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image336.png\" \/><\/p>\n<p>and <img decoding=\"async\" style=\"height: 21px; width: 124px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image333.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 156px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image337.png\" \/><\/p>\n<p>For perpendicular<img decoding=\"async\" style=\"height: 24px; width: 105px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image009.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 27px; width: 169px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image338.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 19px; width: 127px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image339.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image022.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 24px; width: 105px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image009.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 19px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image340.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image017.png\" \/>\u00a0Planes are not perpendicular.<\/p>\n<p>For parallel<img decoding=\"async\" style=\"height: 45px; width: 261px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image341.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image022.png\" \/><img decoding=\"async\" style=\"height: 45px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image328.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/11%20Misc\/image017.png\" \/>\u00a0given planes are parallel.<\/p>\n<p>Therefore, option (B) is correct.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-23\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<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 Miscellaneous 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 &#8230; <a title=\"NCERT Solutions class 12 Maths Miscellaneous\" class=\"read-more\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-9\/\" aria-label=\"More on NCERT Solutions class 12 Maths Miscellaneous\">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-4538","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 Miscellaneous | myCBSEguide<\/title>\n<meta name=\"description\" content=\"NCERT Solutions class 12 Maths Miscellaneous in PDF format for free download. 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