{"id":3649,"date":"2024-01-30T12:24:55","date_gmt":"2024-01-30T06:24:55","guid":{"rendered":"https:\/\/stage-wp.10minuteschool.com\/?p=3649"},"modified":"2024-10-29T16:56:27","modified_gmt":"2024-10-29T10:56:27","slug":"straight-lines-under-conditions","status":"publish","type":"post","link":"https:\/\/10minuteschool.com\/content\/straight-lines-under-conditions\/","title":{"rendered":"\u09ac\u09bf\u09ad\u09bf\u09a8\u09cd\u09a8 \u09b6\u09b0\u09cd\u09a4\u09be\u09a7\u09c0\u09a8\u09c7 \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3"},"content":{"rendered":"<h2><b><span style=\"color: #339966;\">Equation of straight lines under different conditions<\/span><br \/>\n<\/b><\/h2>\n<h3><span style=\"color: #800080;\"><b>\u09a6\u09c1\u0987\u099f\u09bf \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u099b\u09c7\u09a6\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u0997\u09be\u09ae\u09c0 \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 <\/b><b>(The straight line drawn through intersection of two straight lines)\u00a0<\/b><\/span><\/h3>\n<p><img loading=\"lazy\" class=\"aligncenter\" src=\"https:\/\/bl-cms-bkt.s3.amazonaws.com\/prod\/image_91b0c6beb4.png\" alt=\"\u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\" width=\"518\" height=\"270\" \/><\/p>\n<p><b>\u098f\u0995\u099f\u09bf <span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/academic\/10\/\">\u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0<\/a><\/span><\/b> <b>\u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u09a8\u09bf\u09b0\u09cd\u09a3\u09df \u0995\u09b0\u09a4\u09c7 \u09b9\u09ac\u09c7 \u09af\u09be \u09a6\u09c1\u0987\u099f\u09bf<\/b> <b>\u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u099b\u09c7\u09a6\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09a6\u09bf\u09df\u09c7 \u09af\u09be\u09df\u0964<\/b><b>\u00a0\u00a0<\/b><\/p>\n<p><span style=\"font-weight: 400;\">\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09a6\u09cd\u09ac\u09df\u09c7\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3, <span class=\"katex-eq\" data-katex-display=\"false\"> a_{2} x+b_{2} y+c_{2}=0 \\ldots \\quad \\ldots \\quad \\ldots <\/span><\/span><span style=\"font-weight: 400;\">\u00a0 \u00a0(i)<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u098f\u09ac\u0982 <span class=\"katex-eq\" data-katex-display=\"false\"> \\left(a_{1} x+b_{1} y+c_{1}\\right)+k\\left(a_{2} x+b_{2} y+c_{2}\\right)=0 \\ldots \\quad \\ldots <\/span><\/span><span style=\"font-weight: 400;\">\u00a0 (ii)<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09a7\u09b0\u09bf, \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09a6\u09cd\u09ac\u09df\u09c7\u09b0 \u099b\u09c7\u09a6\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09b0 \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995 (<span class=\"katex-eq\" data-katex-display=\"false\"> x_{1},\u00a0 y_{1} <\/span><\/span><span style=\"font-weight: 400;\">) \u09a4\u09be\u09b9\u09b2\u09c7, \u099b\u09c7\u09a6\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09b0 \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995 \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09a6\u09cd\u09ac\u09df\u09c7\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u0995\u09c7 \u09b8\u09bf\u09a6\u09cd\u09a7 \u0995\u09b0\u09ac\u09c7\u0964\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <span class=\"katex-eq\" data-katex-display=\"false\"> a_{1} \\mathrm{x}_{1}+b_{1} y_{1}+c_{1}=0 <\/span>\u00a0 &#8230;.. &#8230;&#8230; &#8230;&#8230; (iii)<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u098f\u09ac\u0982 <\/span><span style=\"font-weight: 400;\">\u00a0<span class=\"katex-eq\" data-katex-display=\"false\"> a_{2} \\mathrm{x}_{1}+b_{2} y_{1}+c_{2}=0 <\/span><\/span><span style=\"font-weight: 400;\">\u2026 \u00a0 \u2026 \u00a0 \u2026\u00a0 (iv)\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u098f\u0996\u09a8 k \u0995\u09c7 \u0987\u099a\u09cd\u099b\u09be\u09a8\u09c1\u09af\u09be\u09df\u09c0 \u09af\u09c7 \u0995\u09cb\u09a8 \u09a7\u09cd\u09b0\u09c1\u09ac\u0995 \u09a7\u09b0\u09c7 (iii) \u0993 (iv) \u09b9\u09a4\u09c7 \u0986\u09ae\u09b0\u09be \u09aa\u09be\u0987,\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<span class=\"katex-eq\" data-katex-display=\"false\"> \\left(a_{1} \\widehat{x}_{1}+b_{1} y_{1}+c_{1}\\right)+\\mathrm{k}\\left(a_{2} \\mathrm{x}_{1}+b_{2} y_{1}+c_{2}\\right)=0 \\ldots \\quad \\ldots \\text { (v) } <\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u0987\u09b9\u09be \u09b8\u09cd\u09aa\u09b7\u09cd\u099f \u09af\u09c7 <span class=\"katex-eq\" data-katex-display=\"false\"> x_{1} \\text{ \u098f\u09ac\u0982 } y_{1} <\/span> <\/span><span style=\"font-weight: 400;\">\u098f\u09b0 \u09ae\u09be\u09a8 \u09a6\u09cd\u09ac\u09be\u09b0\u09be, <span class=\"katex-eq\" data-katex-display=\"false\"> \\left(a_{1} x+b_{1} y+c_{1}\\right)+k\\left(a_{2} x+b_{2} y+c_{2}\\right)=0 \\ldots \\quad \\ldots \\ldots (VI) <\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u099f\u09bf \u09b8\u09bf\u09a6\u09cd\u09a7 \u09b9\u09df, \u09af\u0996\u09a8 k \u09af\u09c7 \u0995\u09cb\u09a8 \u0987\u099a\u09cd\u099b\u09be \u09ae\u09c2\u09b2\u0995 \u09a7\u09cd\u09b0\u09c1\u09ac\u0995 \u098f\u09ac\u0982 <\/span><span style=\"font-weight: 400;\">k\u00a0\u2260\u00a00<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09af\u09c7\u09b9\u09c7\u09a4\u09c1 (vi) \u09a8\u0982 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u099f\u09bf x \u0993 y \u09b8\u09ae\u09cd\u09ac\u09b2\u09bf\u09a4 \u098f\u0995\u0998\u09be\u09a4 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3, \u09b8\u09c1\u09a4\u09b0\u09be\u0982 \u0989\u09b9\u09be (i) \u0993 (ii) \u09a8\u0982 \u09b0\u09c7\u0996\u09be\u09b0 \u099b\u09c7\u09a6\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u0964\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u0985\u09b0\u09cd\u09a5\u09be\u09ce (<span class=\"katex-eq\" data-katex-display=\"false\"> x_{1}, y_{1} <\/span><\/span><span style=\"font-weight: 400;\">) \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09a6\u09bf\u09df\u09c7 \u0985\u09a4\u09bf\u0995\u09cd\u09b0\u09ae\u0995\u09be\u09b0\u09c0 \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u09ac\u09cb\u099d\u09be\u09df\u0964\u00a0<\/span><\/p>\n<p><b>\u09a6\u09c1\u0987\u099f\u09bf <span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/www.youtube.com\/watch?v=cB0DqOSL26M&amp;list=PL1pf33qWCkmidHi8s0NpGydwZAj3fGIhQ&amp;index=1\" target=\"_blank\" rel=\"noopener\">\u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0<\/a><\/span> \u099b\u09c7\u09a6\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u0997\u09be\u09ae\u09c0 \u09a8\u09a4\u09c1\u09a8 \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3, \u098f\u0995 \u09b0\u09c7\u0996\u09be + k (\u0985\u09aa\u09b0 \u09b0\u09c7\u0996\u09be) = 0 \u098f\u0996\u09be\u09a8\u09c7 k \u0990\u099a\u09cd\u099b\u09bf\u0995 \u09a7\u09cd\u09b0\u09c1\u09ac\u0995\u0964\u00a0<\/b><\/p>\n<p><b>\u09a8\u09cb\u099f\u0983<\/b><span style=\"font-weight: 400;\"> k-\u0995\u09c7 \u0987\u099a\u09cd\u099b\u09be\u09ae\u09c2\u09b2\u0995 \u09a7\u09cd\u09b0\u09c1\u09ac\u0995 <\/span><span style=\"font-weight: 400;\">(k\u00a0\u2260\u00a00)\u00a0<\/span><span style=\"font-weight: 400;\">\u09a7\u09b0\u09c7 \u09af\u09a6\u09bf \u098f\u0995\u0997\u09c1\u099a\u09cd\u099b \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3, <span class=\"katex-eq\" data-katex-display=\"false\"> \\left(a_{1} \\times+b_{1} y+c_{1}\\right) + k \\left(a_{2} \\times+b_{2} y+c_{2}\\right) = 0 <\/span><\/span><span style=\"font-weight: 400;\"> \u0986\u0995\u09be\u09b0\u09c7 \u09aa\u09cd\u09b0\u0995\u09be\u09b6 \u0995\u09b0\u09be \u09af\u09be\u09df, \u09a4\u09ac\u09c7 \u0997\u09c1\u099a\u09cd\u099b\u09c7\u09b0 \u09b8\u09ac \u0995\u09df\u099f\u09bf \u09b0\u09c7\u0996\u09be\u0987 \u098f\u0995\u099f\u09bf \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09a6\u09bf\u09df\u09c7 \u09af\u09be\u09ac\u09c7; \u0985\u09b0\u09cd\u09a5\u09be\u09ce <span class=\"katex-eq\" data-katex-display=\"false\"> \\left(a_{1} \\times+b_{1} y+c_{1}\\right) = 0 <\/span><\/span><span style=\"font-weight: 400;\"> \u098f\u09ac\u0982 <span class=\"katex-eq\" data-katex-display=\"false\"> \\left(a_{2} \\times+b_{2} y+c_{2}\\right) = 0 <\/span><\/span><span style=\"font-weight: 400;\">\u00a0\u09b0\u09c7\u0996\u09be\u09a6\u09cd\u09ac\u09df\u09c7\u09b0 \u099b\u09c7\u09a6\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09a6\u09bf\u09df\u09c7 \u09af\u09be\u09ac\u09c7\u0964\u00a0<\/span><\/p>\n<h4><b>\u09e9.\u09e7\u09ec.\u09e8\u00a0 \u098f\u0995\u099f\u09bf \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09ae\u09be\u09a8\u09cd\u09a4\u09b0\u09be\u09b2 \u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u09a8\u09bf\u09b0\u09cd\u09a3\u09df <\/b><b>(Equation of a line parallel to a straight line)<\/b><\/h4>\n<p><span style=\"font-weight: 400;\">\u09a6\u09c1\u0987\u099f\u09bf \u09b8\u09ae\u09be\u09a8\u09cd\u09a4\u09b0\u09be\u09b2 \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u09a2\u09be\u09b2 \u09b8\u09ae\u09be\u09a8 \u09ac\u09b2\u09c7 \u09a4\u09be\u09a6\u09c7\u09b0\u0995\u09c7 <span class=\"katex-eq\" data-katex-display=\"false\"> y=m x+c_{1} <\/span><\/span><span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">\u098f\u09ac\u0982 <span class=\"katex-eq\" data-katex-display=\"false\"> y=m x+c_{2} <\/span><\/span><span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">\u0985\u09a5\u09ac\u09be\u00a0 <span class=\"katex-eq\" data-katex-display=\"false\"> a x+b y+c_{1}=0 <\/span> \u00a0<\/span><span style=\"font-weight: 400;\">\u098f\u09ac\u0982 <span class=\"katex-eq\" data-katex-display=\"false\"> a x+b y+c_{2}=0 <\/span><\/span><span style=\"font-weight: 400;\">\u00a0\u0986\u0995\u09be\u09b0\u09c7 \u09b2\u09c7\u0996\u09be \u09af\u09be\u09df\u0964\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09b8\u09cd\u09aa\u09b7\u09cd\u099f \u09af\u09c7, \u09a6\u09c1\u0987\u099f\u09bf \u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09c7 \u0995\u09c7\u09ac\u09b2 \u09a7\u09cd\u09b0\u09c1\u09ac\u0995 \u09aa\u09a6\u09c7\u09b0 \u09aa\u09be\u09b0\u09cd\u09a5\u0995\u09cd\u09af \u09b9\u09b2\u09c7 \u09a4\u09be\u09b0\u09be \u09aa\u09b0\u09b8\u09cd\u09aa\u09b0 \u09b8\u09ae\u09be\u09a8\u09cd\u09a4\u09b0\u09be\u09b2 \u09b9\u09df\u0964\u00a0\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09b8\u09c1\u09a4\u09b0\u09be\u0982, <span class=\"katex-eq\" data-katex-display=\"false\"> a x+b y+c=0 <\/span> <\/span><span style=\"font-weight: 400;\">\u00a0\u098f\u09b0 \u09b8\u09ae\u09be\u09a8\u09cd\u09a4\u09b0\u09be\u09b2 \u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 <span class=\"katex-eq\" data-katex-display=\"false\"> a x+ b y = k <\/span><\/span><span style=\"font-weight: 400;\">\u00a0\u0986\u0995\u09be\u09b0\u09c7 \u09b2\u09c7\u0996\u09be \u09af\u09be\u09df;\u00a0\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09af\u09c7\u0996\u09be\u09a8\u09c7 k \u098f\u0995\u099f\u09bf \u0987\u099a\u09cd\u099b\u09be\u09ae\u09c2\u09b2\u0995 \u09a7\u09cd\u09b0\u09c1\u09ac\u0995\u0964\u00a0<\/span><\/p>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">\u0986\u09ac\u09be\u09b0, \u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, <span class=\"katex-eq\" data-katex-display=\"false\"> a x+b y+c=0 <\/span> <\/span><span style=\"font-weight: 400;\">\u00a0\u098f\u09b0 \u09b8\u09ae\u09be\u09a8\u09cd\u09a4\u09b0\u09be\u09b2 \u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 <span class=\"katex-eq\" data-katex-display=\"false\"> a x+b y = k <\/span> <\/span><span style=\"font-weight: 400;\">\u00a0\u2026 \u00a0 \u2026 (i)<\/span><\/li>\n<\/ul>\n<p><span style=\"font-weight: 400;\">(i) \u09b0\u09c7\u0996\u09be\u099f\u09bf (<span class=\"katex-eq\" data-katex-display=\"false\"> \\alpha, \\beta <\/span><\/span><span style=\"font-weight: 400;\">) \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u0997\u09be\u09ae\u09c0 \u09b9\u09b2\u09c7 \u0986\u09ae\u09b0\u09be \u09aa\u09be\u0987, <span class=\"katex-eq\" data-katex-display=\"false\"> a \\alpha +b \\beta = k <\/span> <\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u2234<\/span><span style=\"font-weight: 400;\"> (i) \u098f k \u098f\u09b0 \u09ae\u09be\u09a8 \u09ac\u09b8\u09bf\u09df\u09c7 \u09aa\u09be\u0987, <span class=\"katex-eq\" data-katex-display=\"false\"> a x+ b y=a \\alpha + b \\beta <\/span><\/span><\/p>\n<p><b>\u0985\u09a8\u09c1\u09b8\u09bf\u09a6\u09cd\u09a7\u09be\u09a8\u09cd\u09a4\u0983<\/b> <span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\"> a x+b y+c=0 <\/span> <\/span><span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">\u098f\u09b0 \u09b8\u09ae\u09be\u09a8\u09cd\u09a4\u09b0\u09be\u09b2 \u098f\u09ac\u0982 (<span class=\"katex-eq\" data-katex-display=\"false\"> \\alpha, \\beta <\/span>)<\/span><span style=\"font-weight: 400;\"> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u0997\u09be\u09ae\u09c0 \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u00a0 \u00a0 <span class=\"katex-eq\" data-katex-display=\"false\"> a x+ b y=a \\alpha + b \\beta <\/span><\/span><\/p>\n<h4><b>\u09e9.\u09e7\u09ec.\u09e9\u00a0 \u098f\u0995\u099f\u09bf \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u0989\u09aa\u09b0 \u09b2\u09ae\u09cd\u09ac \u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u09a8\u09bf\u09b0\u09cd\u09a3\u09df <\/b><b>(Determine the<\/b><span style=\"font-weight: 400;\">\u00a0<\/span><b>Equation of a perpendicular line to a straight line)<\/b><\/h4>\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\"> y\u00a0=\u00a0mx\u00a0+\u00a0c <\/span> <\/span><span style=\"font-weight: 400;\">\u098f\u09b0 \u09b2\u09ae\u09cd\u09ac \u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u0995\u09be\u09b0\u09a3, <span class=\"katex-eq\" data-katex-display=\"false\"> y=-\\frac{1}{m} x+k <\/span><\/span><span style=\"font-weight: 400;\">; \u09af\u09c7\u0996\u09be\u09a8\u09c7 k \u098f\u0995\u099f\u09bf \u0987\u099a\u09cd\u099b\u09be\u09ae\u09c2\u09b2\u0995 \u09a7\u09cd\u09b0\u09c1\u09ac\u0995\u0964\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u0986\u09ac\u09be\u09b0, <span class=\"katex-eq\" data-katex-display=\"false\"> a x+ b y= c <\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09ac\u09be, <span class=\"katex-eq\" data-katex-display=\"false\"> y=-\\frac{a}{b} x+\\frac{c}{b} <\/span><\/span><span style=\"font-weight: 400;\"> \u098f\u09b0 \u09b2\u09ae\u09cd\u09ac \u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 <span class=\"katex-eq\" data-katex-display=\"false\"> y=\\frac{b}{a} x+k_{1} <\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09ac\u09be, <span class=\"katex-eq\" data-katex-display=\"false\"> b x-a y+a k_{1}=0 <\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\therefore b \\mathrm{x}-a y+k_{1}=0 \\quad \\ldots \\quad \\ldots \\text { (ii)} <\/span> \u09af\u09c7\u0996\u09be\u09a8\u09c7, <span class=\"katex-eq\" data-katex-display=\"false\"> \\mathrm{k}_{1} \\text { \u098f\u09ac\u0982 } k=a k_{1} <\/span> <\/span><span style=\"font-weight: 400;\">\u09a7\u09cd\u09b0\u09c1\u09ac\u0995\u0964\u00a0<\/span><\/p>\n<p><b>\u0985\u09a4\u098f\u09ac, \u09aa\u09cd\u09b0\u09a6\u09a4\u09cd\u09a4 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09c7 x \u0993 y \u098f\u09b0 \u09b8\u09b9\u0997 \u09a6\u09c1\u099f\u09bf \u09aa\u09b0\u09b8\u09cd\u09aa\u09b0 \u09ac\u09bf\u09a8\u09bf\u09ae\u09df \u0995\u09b0\u09c7 \u098f\u09a6\u09c7\u09b0 \u09af\u09c7\u0995\u09cb\u09a8\u09cb \u098f\u0995\u099f\u09bf\u09b0 \u099a\u09bf\u09b9\u09cd\u09a8 \u098f\u09ac\u0982 \u09a7\u09cd\u09b0\u09c1\u09ac \u09aa\u09a6\u09c7\u09b0 \u09aa\u09b0\u09bf\u09ac\u09b0\u09cd\u09a4\u09a8 \u0995\u09b0\u09b2\u09c7 \u0990 \u09b0\u09c7\u0996\u09be\u09b0 \u0989\u09aa\u09b0 \u09b2\u09ae\u09cd\u09ac \u09af\u09c7\u0995\u09cb\u09a8\u09cb \u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u09aa\u09be\u0993\u09df\u09be \u09af\u09be\u09df\u0964\u00a0<\/b><\/p>\n<p><span style=\"font-weight: 400;\">\u09b8\u09c1\u09a4\u09b0\u09be\u0982, <span class=\"katex-eq\" data-katex-display=\"false\"> a x+b y+c=0 <\/span> <\/span><span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">\u09b0\u09c7\u0996\u09be\u09b0 \u0989\u09aa\u09b0 \u09b2\u09ae\u09cd\u09ac\u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 <span class=\"katex-eq\" data-katex-display=\"false\"> b x - a y+k=0 <\/span> <\/span><\/p>\n<p><span style=\"font-weight: 400;\">(ii) \u09b0\u09c7\u0996\u09be\u099f\u09bf (<span class=\"katex-eq\" data-katex-display=\"false\"> \\alpha, \\beta <\/span>)\u00a0 <\/span><span style=\"font-weight: 400;\">\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u0997\u09be\u09ae\u09c0 \u09b9\u09b2\u09c7 \u0986\u09ae\u09b0\u09be \u09aa\u09be\u0987, <span class=\"katex-eq\" data-katex-display=\"false\"> b \\alpha - a \\beta +k= 0 <\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u09ac\u09be, <span class=\"katex-eq\" data-katex-display=\"false\"> \\mathrm{k}=-(b \\alpha-a \\beta) <\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u2234<\/span><span style=\"font-weight: 400;\"> (ii) \u098f k \u098f\u09b0 \u09ae\u09be\u09a8 \u09ac\u09b8\u09bf\u09df\u09c7 \u09aa\u09be\u0987, <span class=\"katex-eq\" data-katex-display=\"false\"> b x-a y-(b \\alpha-a \\beta)=0 <\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u09ac\u09be, <span class=\"katex-eq\" data-katex-display=\"false\"> b x-a y=b \\alpha-a \\beta <\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\therefore a x+b y+c=0 <\/span>\u098f\u09b0 \u0989\u09aa\u09b0 \u09b2\u09ae\u09cd\u09ac \u098f\u09ac\u0982 (<span class=\"katex-eq\" data-katex-display=\"false\"> \\alpha, \\beta <\/span>)\u00a0<\/span><span style=\"font-weight: 400;\"> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u0997\u09be\u09ae\u09c0 \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 <span class=\"katex-eq\" data-katex-display=\"false\"> b x-a y=b \\alpha-a \\beta <\/span><\/span><\/p>\n<h4><b>\u09e9.\u09e7\u09ec.\u09ea\u00a0 \u09a4\u09bf\u09a8\u099f\u09bf \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u098f\u0995\u0987 \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 \u099b\u09c7\u09a6 \u0995\u09b0\u09be\u09b0 \u09a8\u09bf\u09b0\u09cd\u09a3\u09c7\u09df \u09b6\u09b0\u09cd\u09a4 <\/b><b>(The condition that three straight lines are equilateral)<\/b><\/h4>\n<p><span style=\"font-weight: 400;\">\u09a4\u09bf\u09a8\u099f\u09bf \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be <\/span><span style=\"font-weight: 400;\">\u098f\u0995\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7<\/span><span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">\u099b\u09c7\u09a6<\/span><span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">\u0995\u09b0\u09b2\u09c7<\/span><span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">\u09a4\u09be\u09b0\u09be<\/span><span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">\u09b8\u09ae\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1<\/span><span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">\u09b9\u09df<\/span><span style=\"font-weight: 400;\"> \u098f\u09ac\u0982 \u0990 \u09b8\u09be\u09a7\u09be\u09b0\u09a3 \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u0995\u09c7 \u09b8\u09ae\u09cd\u09aa\u09be\u09a4 \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 (Point of concurrence) \u09ac\u09b2\u09be \u09b9\u09df\u0964\u00a0\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, \u09aa\u09cd\u09b0\u09a6\u09a4\u09cd\u09a4 \u09a4\u09bf\u09a8\u099f\u09bf \u09af\u09a5\u09be\u0995\u09cd\u09b0\u09ae\u09c7 <span class=\"katex-eq\" data-katex-display=\"false\"> a_{1} x+b_{1} y+c_{1}=0 \\ldots \\quad \\ldots \\quad \\ldots <\/span><\/span><span style=\"font-weight: 400;\">\u00a0(i)\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <span class=\"katex-eq\" data-katex-display=\"false\"> a_{2} \\mathrm{x}+b_{2} y+c_{2}=0 \\ldots \\quad \\ldots \\quad \\ldots <\/span> <\/span><span style=\"font-weight: 400;\">(ii)\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <span class=\"katex-eq\" data-katex-display=\"false\"> a_{3} \\mathrm{x}+b_{3} y+c_{3}=0 \\ldots \\quad \\ldots \\quad \\ldots <\/span><\/span><span style=\"font-weight: 400;\">\u00a0(iii)\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09af\u09a6\u09bf \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be \u09a4\u09bf\u09a8\u099f\u09bf \u09b8\u09ae\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09b9\u09df, \u09a4\u09be\u09b9\u09b2\u09c7 \u09af\u09c7\u0995\u09cb\u09a8 \u09a6\u09c1\u0987\u099f\u09bf \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u099b\u09c7\u09a6\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09b0 \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995 \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09a4\u09c3\u09a4\u09c0\u09df \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u09b8\u09bf\u09a6\u09cd\u09a7 \u09b9\u09ac\u09c7\u0964\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u098f\u0996\u09a8 \u09ac\u099c\u09cd\u09b0\u0997\u09c1\u09a3\u09a8 \u09aa\u09cd\u09b0\u09a3\u09be\u09b2\u09c0 \u09aa\u09cd\u09b0\u09df\u09cb\u0997 \u0995\u09b0\u09c7 (i) \u0993 (ii) \u09a8\u0982 \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u099b\u09c7\u09a6\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09b0 \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995 \u09b9\u09ac\u09c7,\u00a0<\/span><\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\"> x=\\frac{b_{1} c_{2}-b_{2} c_{1}}{a_{1} b_{2}-a_{2} b_{1}}, y=\\frac{c_{1} a_{2}-c_{2} a_{1}}{a_{1} b_{2}-a_{2} b_{1}} ; a_{1} b_{2}-a_{2} b_{1} \\neq 0 <\/span>\n<p><span style=\"font-weight: 400;\">\u09af\u09a6\u09bf (iii) \u09a8\u0982 \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u099f\u09bf \u098f\u0987 \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09a6\u09bf\u09df\u09c7 \u09af\u09be\u09df, \u09a4\u09be\u09b9\u09b2\u09c7,\u00a0<\/span><\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\"> a_{3}\\left(\\frac{b_{1} c_{2}-b_{2} c_{1}}{a_{1} b_{2}-a_{2} b_{1}}\\right)+b_{3}\\left(\\frac{c_{1} a_{2}-c_{2} a_{1}}{a_{1} b_{2}-a_{2} b_{1}}\\right)+c_{3}=0 <\/span>\n<p><span style=\"font-weight: 400;\">\u09ac\u09be, <span class=\"katex-eq\" data-katex-display=\"false\">\u00a0 a_{3}\\left(b_{1} c_{2}-b_{2} c_{1}\\right)+b_{3}\\left(c_{1} a_{2}-c_{2} a_{1}\\right)+c_{3}\\left(a_{1} b_{2}-a_{2} b_{1}\\right)=0\u00a0<\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\therefore a_{3}\\left(b_{1} c_{2}-b_{2} c_{1}\\right)-b_{3}\\left(c_{2} a_{1}-c_{1} a_{2}\\right)+c_{3}\\left(a_{1} b_{2}-a_{2} b_{1}\\right) <\/span>; \u09af\u09be \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be \u09a4\u09bf\u09a8\u099f\u09bf \u09b8\u09ae\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09b9\u0993\u09df\u09be\u09b0 \u09a8\u09bf\u09b0\u09cd\u09a3\u09c7\u09df \u09b6\u09b0\u09cd\u09a4\u0964\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u0989\u09aa\u09b0\u09bf\u0989\u0995\u09cd\u09a4 \u09b6\u09b0\u09cd\u09a4\u0995\u09c7 \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09df\u0995 <span class=\"katex-eq\" data-katex-display=\"false\"> \\left|\\begin{array}{lll}\n\na_{1} &amp; b_{1} &amp; c_{1} \\\\\n\na_{2} &amp; b_{2} &amp; c_{2} \\\\\n\na_{3} &amp; b_{3} &amp; c_{3}\n\n\\end{array}\\right|=0 <\/span><\/span><span style=\"font-weight: 400;\">\u00a0\u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09aa\u09cd\u09b0\u0995\u09be\u09b6 \u0995\u09b0\u09be \u09af\u09be\u09df\u0964\u00a0<\/span><\/p>\n<p><b>\u09a8\u09cb\u099f\u0983<\/b> <span style=\"font-weight: 400;\">\u09a4\u09bf\u09a8\u099f\u09bf \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be \u09a6\u09c7\u0993\u09df\u09be \u09a5\u09be\u0995\u09b2\u09c7, \u09b0\u09c7\u0996\u09be \u09a4\u09bf\u09a8\u099f\u09bf \u09b8\u09ae\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09b9\u0993\u09df\u09be\u09b0 \u09b6\u09b0\u09cd\u09a4 \u09a4\u09be\u09a6\u09c7\u09b0 \u09b8\u09b9\u0997\u0997\u09c1\u09b2\u09cb\u09b0 \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09df\u0995\u09c7\u09b0 \u09ae\u09be\u09a8 \u09b6\u09c1\u09a8\u09cd\u09af \u09b9\u09ac\u09c7\u0964\u00a0<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Equation of straight lines under different conditions \u09a6\u09c1\u0987\u099f\u09bf \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u099b\u09c7\u09a6\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u0997\u09be\u09ae\u09c0 \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 (The straight line drawn through intersection of two straight lines)\u00a0 \u098f\u0995\u099f\u09bf \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u09a8\u09bf\u09b0\u09cd\u09a3\u09df \u0995\u09b0\u09a4\u09c7 \u09b9\u09ac\u09c7 \u09af\u09be \u09a6\u09c1\u0987\u099f\u09bf \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u099b\u09c7\u09a6\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09a6\u09bf\u09df\u09c7 \u09af\u09be\u09df\u0964\u00a0\u00a0 \u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09a6\u09cd\u09ac\u09df\u09c7\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3,<\/p>\n<p> <a class=\"redmore\" href=\"https:\/\/10minuteschool.com\/content\/straight-lines-under-conditions\/\">Read More<\/a><\/p>\n","protected":false},"author":56,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[4258,3037,3026],"tags":[2534,2531,2533,2532],"_links":{"self":[{"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/posts\/3649"}],"collection":[{"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/users\/56"}],"replies":[{"embeddable":true,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/comments?post=3649"}],"version-history":[{"count":16,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/posts\/3649\/revisions"}],"predecessor-version":[{"id":16100,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/posts\/3649\/revisions\/16100"}],"wp:attachment":[{"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/media?parent=3649"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/categories?post=3649"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/tags?post=3649"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}