{"id":3648,"date":"2024-01-30T12:25:03","date_gmt":"2024-01-30T06:25:03","guid":{"rendered":"https:\/\/stage-wp.10minuteschool.com\/?p=3648"},"modified":"2024-10-29T16:32:13","modified_gmt":"2024-10-29T10:32:13","slug":"definite-straight-line","status":"publish","type":"post","link":"https:\/\/10minuteschool.com\/content\/definite-straight-line\/","title":{"rendered":"\u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09be\u09aa\u09c7\u0995\u09cd\u09b7\u09c7 \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u0993 \u09b0\u09c7\u0996\u09be\u0982\u09b6\u09c7\u09b0 \u09aa\u09cd\u09b0\u09a4\u09bf\u099a\u09cd\u099b\u09ac\u09bf"},"content":{"rendered":"<h2><span style=\"color: #339966;\"><b>\u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09be\u09aa\u09c7\u0995\u09cd\u09b7\u09c7 \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u0993 \u09b0\u09c7\u0996\u09be\u0982\u09b6\u09c7\u09b0 \u09aa\u09cd\u09b0\u09a4\u09bf\u099a\u09cd\u099b\u09ac\u09bf (<\/b><b>Image of a point and a straight line with respect to a definite straight line)<\/b><\/span><\/h2>\n<p><span style=\"font-weight: 400;\">\u0986\u09ae\u09bf \u09af\u09a6\u09bf \u0986\u09af\u09bc\u09a8\u09be\u09b0 \u09b8\u09be\u09ae\u09a8\u09c7 \u098f\u0995\u099f\u09bf \u0995\u09b2\u09ae\u0995\u09c7 \u09a7\u09b0\u09bf \u09b8\u09c7\u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7 \u09b2\u0995\u09cd\u09b7\u09cd\u09af \u0995\u09b0\u09be \u09af\u09be\u09af\u09bc \u09af\u09c7, \u0986\u09af\u09bc\u09a8\u09be\u09b0 \u09aa\u09c3\u09b7\u09cd\u09a0 \u09a5\u09c7\u0995\u09c7 \u09a0\u09bf\u0995 \u09af\u09a4 \u09a6\u09c2\u09b0\u09c7 \u0995\u09b2\u09ae\u099f\u09bf \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09a8 \u0995\u09b0\u099b\u09c7 \u0995\u09b2\u09ae\u09c7\u09b0 \u09aa\u09cd\u09b0\u09a4\u09bf\u099a\u09cd\u099b\u09ac\u09bf\u0993 \u0986\u09af\u09bc\u09a8\u09be\u09b0 \u09aa\u09c3\u09b7\u09cd\u09a0 \u09a5\u09c7\u0995\u09c7 \u09a0\u09bf\u0995 \u09a4\u09a4\u099f\u09be \u09ad\u09bf\u09a4\u09b0\u09c7 \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09a8 \u0995\u09b0\u099b\u09c7 (\u098f\u099f\u09bf \u09aa\u09b0\u09bf\u09ae\u09be\u09aa\u09af\u09cb\u0997\u09cd\u09af \u09a8\u09be \u09b9\u09b2\u09c7\u0993 \u09a6\u09c3\u09b7\u09cd\u099f\u09bf \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09ac\u09c1\u099d\u09be \u09af\u09be\u09af\u09bc)\u0964 \u098f\u0996\u09a8 \u0995\u09b2\u09ae\u09c7\u09b0 \u09aa\u09b0\u09bf\u09ac\u09b0\u09cd\u09a4\u09c7 \u098f\u0995\u099f\u09bf \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u098f\u09ac\u0982 \u0986\u09af\u09bc\u09a8\u09be\u09b0 \u09aa\u09c3\u09b7\u09cd\u09a0 \u09b8\u09ae\u09a4\u09b2\u09c7 \u098f\u0995\u099f\u09bf \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09be\u09aa\u09c7\u0995\u09cd\u09b7\u09c7 \u09ac\u09bf\u09b7\u09af\u09bc\u099f\u09bf \u09ac\u09bf\u09ac\u09c7\u099a\u09a8\u09be \u0995\u09b0\u09b2\u09c7\u0993 \u098f\u0995\u0987 \u09b9\u09ac\u09c7\u0964<\/span><\/p>\n<p><b>\u201c\u098f\u0995\u099f\u09bf \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09be\u09aa\u09c7\u0995\u09cd\u09b7\u09c7 \u0995\u09cb\u09a8\u09cb \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09b0 \u09aa\u09cd\u09b0\u09a4\u09bf\u099a\u09cd\u099b\u09ac\u09bf \u09a8\u09bf\u09b0\u09cd\u09a3\u09af\u09bc\u09c7 \u0995\u09b0\u09a4\u09c7 \u09b9\u09b2\u09c7 \u09b0\u09c7\u0996\u09be \u09ac\u09b0\u09be\u09ac\u09b0 \u0995\u09be\u0997\u099c\u099f\u09bf \u09ad\u09be\u0981\u099c \u0995\u09b0\u09b2\u09c7 \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u099f\u09bf \u0995\u09be\u0997\u099c\u09c7\u09b0 \u0985\u09aa\u09b0 \u09aa\u09be\u09b0\u09cd\u09b6\u09cd\u09ac\u09c7 \u09af\u09c7 \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 \u09b8\u09cd\u09aa\u09b0\u09cd\u09b6 \u0995\u09b0\u09ac\u09c7 \u09b8\u09c7 \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u0987 \u09aa\u09cd\u09b0\u09a5\u09ae \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09b0 \u09aa\u09cd\u09b0\u09a4\u09bf\u099a\u09cd\u099b\u09ac\u09bf\u0964\u201d<\/b><\/p>\n<p><span style=\"font-weight: 400;\">\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, <span class=\"katex-eq\" data-katex-display=\"false\"> a x+b y+c=0 \\text { (i) } <\/span><\/span><span style=\"font-weight: 400;\">\u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09be\u09aa\u09c7\u0995\u09cd\u09b7\u09c7 <span class=\"katex-eq\" data-katex-display=\"false\"> P \\left(x_{1}, y_{1}\\right) <\/span><\/span><span style=\"font-weight: 400;\"> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09b0 \u09aa\u09cd\u09b0\u09a4\u09bf\u099a\u09cd\u099b\u09ac\u09bf <span class=\"katex-eq\" data-katex-display=\"false\"> P^{\\prime}\\left(x_{2}, y_{2}\\right) <\/span><\/span><span style=\"font-weight: 400;\">\u00a0\u09a8\u09bf\u09b0\u09cd\u09a3\u09af\u09bc \u0995\u09b0\u09a4\u09c7 \u09b9\u09ac\u09c7\u0964<\/span><\/p>\n<p><span style=\"font-weight: 400;\">(i) \u09a8\u0982 \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u09a2\u09be\u09b2 = <span class=\"katex-eq\" data-katex-display=\"false\"> - \\frac{a}{b} <\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u098f\u099f\u09be \u09b8\u09cd\u09aa\u09b7\u09cd\u099f \u09af\u09c7, PP<\/span><span style=\"font-weight: 400;\">&#8216;<\/span><span style=\"font-weight: 400;\"> \u09b0\u09c7\u0996\u09be (i) \u09a8\u0982 \u09b0\u09c7\u0996\u09be\u09b0 \u0989\u09aa\u09b0 \u09b2\u09ae\u09cd\u09ac \u098f\u09ac\u0982 R \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 PP\u2019 \u098f\u09b0 \u09ae\u09a7\u09cd\u09af\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u0964 \u09b8\u09c1\u09a4\u09b0\u09be\u0982 PP<\/span><span style=\"font-weight: 400;\">\u2019<\/span><span style=\"font-weight: 400;\"> \u09b0\u09c7\u0996\u09be\u09b0 \u09a2\u09be\u09b2 = <\/span><span style=\"font-weight: 400;\"> <span class=\"katex-eq\" data-katex-display=\"false\"> \\frac{a}{b} <\/span><\/span><span style=\"font-weight: 400;\">: \u09b8\u09c1\u09a4\u09b0\u09be\u0982 PP<\/span><span style=\"font-weight: 400;\">\u2019<\/span><span style=\"font-weight: 400;\"> \u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u00a0 <span class=\"katex-eq\" data-katex-display=\"false\"> y-y_{1}=\\frac{b}{a}\\left(x-x_{1}\\right) <\/span><\/span><span style=\"font-weight: 400;\"> \u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, R \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09b0 \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995 (<span class=\"katex-eq\" data-katex-display=\"false\"> \\alpha, \\beta <\/span><\/span><span style=\"font-weight: 400;\">) [ R \u09b9\u09b2\u09cb (i) \u09a8\u0982 \u0993 PP<\/span><span style=\"font-weight: 400;\">\u2019<\/span><span style=\"font-weight: 400;\"> \u09b0\u09c7\u0996\u09be\u09b0 \u099b\u09c7\u09a6 \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 ]<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09b8\u09c1\u09a4\u09b0\u09be\u0982, <span class=\"katex-eq\" data-katex-display=\"false\"> \\left(\\frac{x_{1}+x_{2}}{2}, \\frac{y_{1}+y_{2}}{2}\\right)=(\\alpha, \\beta) <\/span><\/span><\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\"> \\therefore \\frac{x_{1}+x_{2}}{2}=\\alpha \\quad \\text { \u098f\u09ac\u0982, } \\frac{y_{1}+y_{2}}{2}=\\beta <\/span>\n<span class=\"katex-eq\" data-katex-display=\"false\"> \\therefore x_{2}=2 \\alpha-x_{1} \\quad \\text { \u09ac\u09be, } y_{2}=2 \\beta-y_{1} <\/span>\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\therefore \\mathrm{P}\\left(\\mathrm{x}_{1}, \\mathrm{y}_{1}\\right) <\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09b0 \u09aa\u09cd\u09b0\u09a4\u09bf\u099a\u09cd\u099b\u09ac\u09bf <span class=\"katex-eq\" data-katex-display=\"false\"> \\left(2 \\alpha-x_{1}, 2 \\beta-y_{1}\\right) <\/span><\/span><\/p>\n<p><b>\u09a6\u09cd\u09b0\u09b7\u09cd\u099f\u09ac\u09cd\u09af : <\/b><b>\u09e7. <\/b><span style=\"font-weight: 400;\">\u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09be\u09aa\u09c7\u0995\u09cd\u09b7\u09c7 \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09b0 \u09aa\u09cd\u09b0\u09a4\u09bf\u099a\u09cd\u099b\u09ac\u09bf \u09a8\u09bf\u09b0\u09cd\u09a3\u09af\u09bc\u09c7\u09b0 \u09b8\u09ae\u09af\u09bc \u09ae\u09a8\u09c7 \u09b0\u09be\u0996\u09a4\u09c7 \u09b9\u09ac\u09c7\u0964<\/span><\/p>\n<p><b>\u09e8. <\/b><span style=\"font-weight: 400;\">\u0995\u09cb\u09a8\u09cb \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u0993 \u09a4\u09be\u09b0 \u09aa\u09cd\u09b0\u09a4\u09bf\u099a\u09cd\u099b\u09ac\u09bf \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09b8\u09b0\u09cd\u09ac\u09a6\u09be\u0987 \u09aa\u09cd\u09b0\u09a4\u09bf\u09ab\u09b2\u09a8 \u09b0\u09c7\u0996\u09be \u09a5\u09c7\u0995\u09c7 \u09b8\u09ae\u09a6\u09c2\u09b0\u09ac\u09b0\u09cd\u09a4\u09c0\u0964<\/span><\/p>\n<p><b>\u09e9. <\/b><span style=\"font-weight: 400;\">\u0995\u09cb\u09a8\u09cb \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u0993 \u09a4\u09be\u09b0 \u09aa\u09cd\u09b0\u09a4\u09bf\u099a\u09cd\u099b\u09ac\u09bf \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09ac\u09b0\u09be\u09ac\u09b0 \u09b8\u0982\u09af\u09cb\u0997 \u09b0\u09c7\u0996\u09be \u09b8\u09b0\u09cd\u09ac\u09a6\u09be\u0987 \u09aa\u09cd\u09b0\u09a4\u09bf\u09ab\u09b2\u09a8 \u09b0\u09c7\u0996\u09be\u09b0 \u0989\u09aa\u09b0 \u09b2\u09ae\u09cd\u09ac \u0964<\/span><\/p>\n<p><b>\u0995\u09cb\u09a8\u09cb<\/b> <b>\u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09be\u09a5\u09c7 \u09b0\u09c7\u0996\u09be\u0982\u09b6\u09c7\u09b0 \u09aa\u09cd\u09b0\u09a4\u09bf\u099a\u09cd\u099b\u09ac\u09bf : <\/b><span style=\"font-weight: 400;\">\u098f\u0995\u099f\u09bf \u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09be\u09aa\u09c7\u0995\u09cd\u09b7\u09c7 \u0986\u09b0\u09c7\u0995\u099f\u09bf \u09b0\u09c7\u0996\u09be\u09b0 \u09aa\u09cd\u09b0\u09a4\u09bf\u099a\u09cd\u099b\u09ac\u09bf \u098f\u09ae\u09a8 \u098f\u0995 \u09a7\u09b0\u09a8\u09c7\u09b0 \u09b0\u09c2\u09aa\u09be\u09a8\u09cd\u09a4\u09b0 \u09af\u09be\u09a4\u09c7 \u098f\u0995\u099f\u09bf \u09b0\u09c7\u0996\u09be\u09af\u09bc \u0985\u09ac\u09b8\u09cd\u09a5\u09bf\u09a4 \u09aa\u09cd\u09b0\u09a4\u09bf\u099f\u09bf \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09b0 \u09aa\u09cd\u09b0\u09a4\u09bf\u099a\u09cd\u099b\u09ac\u09bf \u098f\u09ac\u0982 \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u0997\u09c1\u09b2\u09cb \u09aa\u09b0\u09b8\u09cd\u09aa\u09b0 \u09b2\u09ae\u09cd\u09ac\u09ad\u09be\u09ac\u09c7 \u09aa\u09cd\u09b0\u09a4\u09bf\u09ab\u09b2\u09a8 \u09b0\u09c7\u0996\u09be \u09b9\u09a4\u09c7 \u09b8\u09ae\u09a6\u09c2\u09b0\u09ac\u09b0\u09cd\u09a4\u09c0\u0964 \u09aa\u09cd\u09b0\u09a4\u09bf\u09ab\u09b2\u0995 \u09b0\u09c7\u0996\u09be \u09aa\u09cd\u09b0\u09a4\u09bf\u099f\u09bf \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u0993 \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09b0 \u09aa\u09cd\u09b0\u09a4\u09bf\u099a\u09cd\u099b\u09ac\u09bf\u09b0 \u09b8\u0982\u09af\u09cb\u099c\u0995 \u09b0\u09c7\u0996\u09be\u09b0 \u09b2\u09ae\u09cd\u09ac\u09a6\u09cd\u09ac\u09bf\u0996\u09a3\u09cd\u09a1\u0995\u0964 \u09b0\u09c7\u0996\u09be\u09a6\u09cd\u09ac\u09af\u09bc\u09c7\u09b0 \u099b\u09c7\u09a6\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u0993 \u09ae\u09a7\u09cd\u09af\u09ac\u09b0\u09cd\u09a4\u09c0 \u0995\u09cb\u09a3 \u09b9\u09a4\u09c7<\/span> <span style=\"font-weight: 400;\">\u0995\u09cb\u09a8\u09cb<\/span><span style=\"font-weight: 400;\"> \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09b0\u09c7\u0996\u09be\u09b0 \u09aa\u09cd\u09b0\u09a4\u09bf\u099a\u09cd\u099b\u09ac\u09bf\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u09aa\u09be\u0993\u09af\u09bc\u09be \u09af\u09be\u09af\u09bc \u0964<\/span><\/p>\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\"> L_{1} \\equiv a x+b y+c=0 <\/span> \u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09be\u09aa\u09c7\u0995\u09cd\u09b7\u09c7 <span class=\"katex-eq\" data-katex-display=\"false\"> L_{2} \\equiv a_{1} x+b_{1} y+c_{1}=0 <\/span><\/span><span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">\u09b0\u09c7\u0996\u09be\u09b0 \u09aa\u09cd\u09b0\u09a4\u09bf\u099a\u09cd\u099b\u09ac\u09bf \u09a8\u09bf\u09b0\u09cd\u09a3\u09af\u09bc \u0995\u09b0\u09a4\u09c7 \u09b9\u09ac\u09c7\u0964<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, <span class=\"katex-eq\" data-katex-display=\"false\"> L_{1} <\/span><\/span><span style=\"font-weight: 400;\"> \u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09be\u09aa\u09c7\u0995\u09cd\u09b7\u09c7 <span class=\"katex-eq\" data-katex-display=\"false\"> L_{2} <\/span>\u00a0<\/span><span style=\"font-weight: 400;\"> \u09b0\u09c7\u0996\u09be\u09b0 \u09aa\u09cd\u09b0\u09a4\u09bf\u099a\u09cd\u099b\u09ac\u09bf <span class=\"katex-eq\" data-katex-display=\"false\"> L_{3} \\equiv a_{2} x+b_{2} y+c_{2}=0\/katex] &lt;\/span&gt;&lt;span style=&quot;font-weight: 400;&quot;&gt;\u098f\u09ac\u0982 [katex] \\mathrm{L}_{1}, \\mathrm{~L}_{2} <\/span><\/span><span style=\"font-weight: 400;\"> \u0993 <span class=\"katex-eq\" data-katex-display=\"false\"> L_{3} <\/span> <\/span><span style=\"font-weight: 400;\">\u00a0\u09b0\u09c7\u0996\u09be\u09b0 \u09a2\u09be\u09b2 \u09af\u09a5\u09be\u0995\u09cd\u09b0\u09ae\u09c7 <span class=\"katex-eq\" data-katex-display=\"false\"> \\mathrm{m}_{1}, \\mathrm{~m}_{2} \\text { \u0993 } \\mathrm{m}_{3} <\/span><\/span><\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\"> \\therefore \\tan \\varphi_{12}=\\frac{m_{2}-m_{1}}{1+m_{1} m_{2}} \\ldots \\quad \\ldots \\quad \\ldots \\text { (i) } <\/span>\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\therefore L_{2} <\/span> \u09b0\u09c7\u0996\u09be\u09b0 \u09aa\u09cd\u09b0\u09a4\u09bf\u099a\u09cd\u099b\u09ac\u09bf <span class=\"katex-eq\" data-katex-display=\"false\"> L_{3} <\/span> \u0993 <span class=\"katex-eq\" data-katex-display=\"false\"> L_{1} <\/span>\u00a0 \u09b0\u09c7\u0996\u09be\u09b0 \u09ae\u09a7\u09cd\u09af\u09ac\u09b0\u09cd\u09a4\u09c0 \u0995\u09cb\u09a3 <span class=\"katex-eq\" data-katex-display=\"false\"> \\pi-\\varphi_{12} <\/span><\/span><\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\"> \\therefore \\tan \\varphi\\left(\\pi-\\varphi_{12}\\right)=\\frac{m_{3}-m_{1}}{1+m_{1} m_{3}} <\/span>\n<p><span style=\"font-weight: 400;\">\u09ac\u09be, <span class=\"katex-eq\" data-katex-display=\"false\"> -\\tan \\varphi_{12}=\\frac{m_{3}-m_{1}}{1+m_{1} m_{3}} \\ldots \\quad \\ldots \\quad \\ldots \\text { (ii) } <\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">(i) \u0993 (ii) \u09a8\u0982 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u09b9\u09a4\u09c7 \u09aa\u09be\u0987,<\/span><\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\"> -\\frac{m_{2}-m_{1}}{1+m_{1} m_{2}}=\\frac{m_{3}-m_{1}}{1+m_{1} m_{3}} \\ldots \\quad \\ldots \\quad \\ldots \\text { (iii) } <\/span>\n<p><span style=\"font-weight: 400;\">iii) \u09a8\u0982 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u09b8\u09ae\u09be\u09a7\u09be\u09a8 \u0995\u09b0\u09b2\u09c7 <span class=\"katex-eq\" data-katex-display=\"false\"> m_{3} <\/span><\/span><span style=\"font-weight: 400;\">\u00a0\u098f\u09b0 \u09ae\u09be\u09a8 \u09aa\u09be\u0993\u09af\u09bc\u09be \u09af\u09be\u09ac\u09c7\u0964<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u0986\u09ac\u09be\u09b0, <span class=\"katex-eq\" data-katex-display=\"false\"> L_{1}, L_{2} <\/span><\/span><span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">\u09b0\u09c7\u0996\u09be\u09b0 \u099b\u09c7\u09a6\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 (<span class=\"katex-eq\" data-katex-display=\"false\"> x_{1}, y_{1} <\/span><\/span><span style=\"font-weight: 400;\">) \u09b9\u09b2\u09c7 <span class=\"katex-eq\" data-katex-display=\"false\"> L_{3} <\/span> <\/span><span style=\"font-weight: 400;\">\u00a0\u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3,<\/span><\/p>\n<p><span class=\"katex-eq\" data-katex-display=\"false\"> y-y_{1}=m_{3}\\left(x-x_{1}\\right) \\ldots \\ldots \\ldots \\text { (iv) } <\/span><b>\u00a0 \u00a0 <\/b><span style=\"font-weight: 400;\">[\u09af\u09c7\u09b9\u09c7\u09a4\u09c1 <span class=\"katex-eq\" data-katex-display=\"false\"> L_{1} L_{2} \\text { \u0993 } L_{3} <\/span><\/span><span style=\"font-weight: 400;\">\u09b8\u09ae\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1]<\/span><\/p>\n<p><b>\u0989\u09a6\u09be\u09b9\u09b0\u09a3 : <\/b><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\"> 2 x-y+1=0 <\/span> <\/span><b><a href=\"https:\/\/10minuteschool.com\/academic\/10\/\"><span style=\"color: #0000ff;\">\u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be<\/span><\/a>\u09b0 \u09b8\u09be\u09aa\u09c7\u0995\u09cd\u09b7\u09c7 <\/b><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\"> x+y-2=0 <\/span> <\/span><b><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<\/a><\/span>\u09b0 \u09aa\u09cd\u09b0\u09a4\u09bf\u099a\u09cd\u099b\u09ac\u09bf\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u09a8\u09bf\u09b0\u09cd\u09a3\u09af\u09bc \u0995\u09b0\u0964<\/b><\/p>\n<p><b>\u09b8\u09ae\u09be\u09a7\u09be\u09a8 <\/b><b>: <\/b><span style=\"font-weight: 400;\">\u09a7\u09b0\u09bf, <span class=\"katex-eq\" data-katex-display=\"false\"> L_{1} \\equiv 2 x-y+z=0 <\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u098f\u09ac\u0982, <span class=\"katex-eq\" data-katex-display=\"false\"> L_{2} \\equiv x+y-2=0 <\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09a7\u09b0\u09bf, <span class=\"katex-eq\" data-katex-display=\"false\"> \\mathrm{L}_{1} <\/span><\/span><span style=\"font-weight: 400;\"> \u0993 <span class=\"katex-eq\" data-katex-display=\"false\"> \\mathrm{L}_{2} <\/span><\/span><span style=\"font-weight: 400;\">\u00a0\u09b0\u09c7\u0996\u09be\u09b0 \u09ae\u09a7\u09cd\u09af\u09ac\u09b0\u09cd\u09a4\u09c0 \u0995\u09cb\u09a3 <\/span><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\varphi_{12} <\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\"> L_{1} \\text{ \u0993 } L_{2} <\/span> \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09a6\u09cd\u09ac\u09df\u09c7\u09b0 \u09a2\u09be\u09b2 \u09af\u09a5\u09be\u0995\u09cd\u09b0\u09ae\u09c7,<\/span><\/p>\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\"> m_{1}=2 <\/span> \u0993 <span class=\"katex-eq\" data-katex-display=\"false\"> m_{2}=-1 <\/span><\/span><\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\"> \\therefore \\tan \\varphi_{12}=\\frac{-1-2}{1+(-1) \\cdot 2}=3 <\/span>\n<p><span style=\"font-weight: 400;\">\u0986\u09ac\u09be\u09b0, <span class=\"katex-eq\" data-katex-display=\"false\"> L_{1} \\text{ \u0993 } L_{3} <\/span><\/span><span style=\"font-weight: 400;\"> \u09b0\u09c7\u0996\u09be\u09a6\u09cd\u09ac\u09af\u09bc\u09c7\u09b0 \u09ae\u09a7\u09cd\u09af\u09ac\u09b0\u09cd\u09a4\u09c0 \u0995\u09cb\u09a3 <\/span><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\pi-\\varphi_{12} <\/span><\/span><span style=\"font-weight: 400;\"> \u0993 <span class=\"katex-eq\" data-katex-display=\"false\"> L_{3} <\/span> <\/span><span style=\"font-weight: 400;\">\u00a0\u09b0\u09c7\u0996\u09be\u09b0 \u09a2\u09be\u09b2 <span class=\"katex-eq\" data-katex-display=\"false\"> m_{3} <\/span><\/span><span style=\"font-weight: 400;\">\u00a0\u09b9\u09b2\u09c7,<\/span><\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\"> \\tan \\left(\\pi-\\varphi_{12}\\right)=\\frac{m_{3}-2}{1+2 m_{3}} <\/span>\n<p><span style=\"font-weight: 400;\">\u09ac\u09be, <span class=\"katex-eq\" data-katex-display=\"false\"> -\\tan \\varphi_{12}=\\frac{m_{3}-2}{1+2 m_{3}} <\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09ac\u09be, <span class=\"katex-eq\" data-katex-display=\"false\"> -3=\\frac{m_{3}-2}{1+2 m_{3}} <\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09ac\u09be, <span class=\"katex-eq\" data-katex-display=\"false\"> -3-6 m_{3}=2 m_{3} <\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09ac\u09be, <span class=\"katex-eq\" data-katex-display=\"false\"> 7 m_{3}=-3+2 <\/span><\/span><\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\"> \\therefore m_{3}=\\frac{-1}{7} <\/span>\n<p><span style=\"font-weight: 400;\">\u0986\u09ac\u09be\u09b0, \u09ac\u099c\u09cd\u09b0\u0997\u09c1\u09a3\u09a8 \u09aa\u09a6\u09cd\u09a7\u09a4\u09bf\u09a4\u09c7 \u09aa\u09be\u0987 <span class=\"katex-eq\" data-katex-display=\"false\"> L_{1} <\/span> <\/span><span style=\"font-weight: 400;\">\u00a0\u0993 <span class=\"katex-eq\" data-katex-display=\"false\"> L_{2} <\/span>\u00a0<\/span><span style=\"font-weight: 400;\"> \u09b0\u09c7\u0996\u09be\u09a6\u09cd\u09ac\u09af\u09bc\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\"> \\left(\\frac{1}{3}, \\frac{5}{3}\\right) <\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\therefore L_{1} <\/span>\u00a0 \u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09be\u09aa\u09c7\u0995\u09cd\u09b7\u09c7 <span class=\"katex-eq\" data-katex-display=\"false\"> L_{2} <\/span>\u00a0<\/span><span style=\"font-weight: 400;\">\u00a0\u09b0\u09c7\u0996\u09be\u09b0 \u09aa\u09cd\u09b0\u09a4\u09bf\u099a\u09cd\u099b\u09ac\u09bf\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3,<\/span><\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\"> y-\\frac{5}{3}=-\\frac{5}{3}\\left(x-\\frac{1}{3}\\right) <\/span>\n<p><span style=\"font-weight: 400;\">\u09ac\u09be, <span class=\"katex-eq\" data-katex-display=\"false\"> 3 y-5=-\\frac{1}{7}(3 x-1) <\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09ac\u09be, <\/span><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\"> 21 y-35=-3 x+1<\/span><\/span><\/p>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><b>\u098f\u0995\u099f\u09bf \u09a4\u09cd\u09b0\u09bf\u09ad\u09c1\u099c\u09c7\u09b0 \u09b6\u09c0\u09b0\u09cd\u09b7\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09a4\u09bf\u09a8\u099f\u09bf <span class=\"katex-eq\" data-katex-display=\"false\"> A\\left(x_{1}, y_{1}\\right), B\\left(x_{2}, y_{2}\\right), C\\left(x_{3}, y_{3}\\right) <\/span><\/b><b>\u00a0\u09b9\u09b2\u09c7 \u09a4\u09cd\u09b0\u09bf\u09ad\u09c1\u099c\u099f\u09bf\u09b0 \u0985\u09a8\u09cd\u09a4\u0983\u0995\u09c7\u09a8\u09cd\u09a6\u09cd\u09b0 \u09a8\u09bf\u09b0\u09cd\u09a3\u09af\u09bc \u098f\u0995\u099f\u09bf \u09a4\u09cd\u09b0\u09bf\u09ad\u09c1\u099c\u09c7\u09b0 \u09a4\u09bf\u09a8\u099f\u09bf \u0995\u09cb\u09a3\u09c7\u09b0 \u09b8\u09ae\u09a4\u09cd\u09b0\u09bf\u0996\u09a3\u09cd\u09a1\u0995 \u09af\u09c7 \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 \u09ae\u09bf\u09b2\u09bf\u09a4 \u09b9\u09af\u09bc, \u09a4\u09be\u0995\u09c7 \u0985\u09a8\u09cd\u09a4\u0983\u0995\u09c7\u09a8\u09cd\u09a6\u09cd\u09b0 \u09ac\u09b2\u09c7\u0964<\/b><\/li>\n<\/ul>\n<p><span style=\"font-weight: 400;\">\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, <span class=\"katex-eq\" data-katex-display=\"false\"> \\Delta A B C <\/span><\/span><span style=\"font-weight: 400;\"> \u098f\u09b0 \u09ac\u09be\u09b9\u09c1\u0997\u09c1\u09b2\u09cb\u09b0 \u09a6\u09c8\u09b0\u09cd\u0998\u09cd\u09af BC= a, CA = b \u098f\u09ac\u0982 AB = c. <span class=\"katex-eq\" data-katex-display=\"false\"> \\angle \\mathrm{A}, \\angle \\mathrm{B} \\text{ \u0993} \\angle \\mathrm{C} <\/span><\/span><span style=\"font-weight: 400;\">\u098f\u09b0 \u09b8\u09ae\u09a6\u09cd\u09ac\u09bf\u0996\u09a3\u09cd\u09a1\u0995 AI, BI \u0993 CI \u09aa\u09b0\u09b8\u09cd\u09aa\u09b0 I \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 \u099b\u09c7\u09a6 \u0995\u09b0\u09c7\u0964 \u09b8\u09c1\u09a4\u09b0\u09be\u0982 I \u09a4\u09cd\u09b0\u09bf\u09ad\u09c1\u099c\u09c7\u09b0 \u0985\u09a8\u09cd\u09a4\u0983\u0995\u09c7\u09a8\u09cd\u09a6\u09cd\u09b0\u0964<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u098f\u0996\u09a8 AI \u0995\u09c7 D \u09aa\u09b0\u09cd\u09af\u09a8\u09cd\u09a4 \u098f\u09ae\u09a8\u09ad\u09be\u09ac\u09c7 \u09ac\u09b0\u09cd\u09a7\u09bf\u09a4 \u0995\u09b0\u09bf \u09af\u09c7\u09a8 BC \u0995\u09c7 D \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 \u099b\u09c7\u09a6 \u0995\u09b0\u09c7\u0964<\/span><\/p>\n<p><b>\u0986\u09ae\u09b0\u09be \u099c\u09be\u09a8\u09bf, \u098f\u0995\u099f\u09bf \u09a4\u09cd\u09b0\u09bf\u09ad\u09c1\u099c\u09c7\u09b0 \u09b6\u09c0\u09b0\u09cd\u09b7 \u0995\u09cb\u09a3\u09c7\u09b0 \u09b8\u09ae\u09a6\u09cd\u09ac\u09bf\u0996\u09a3\u09cd\u09a1\u0995 \u09b0\u09c7\u0996\u09be \u09a4\u09be\u09b0 \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u09ac\u09be\u09b9\u09c1\u0995\u09c7 \u09af\u09c7 \u0985\u09a8\u09c1\u09aa\u09be\u09a4\u09c7 \u099b\u09c7\u09a6 \u0995\u09b0\u09c7 \u09a4\u09be \u09b8\u09a8\u09cd\u09a8\u09bf\u09b9\u09bf\u09a4 \u09ac\u09be\u09b9\u09c1\u09a6\u09cd\u09ac\u09af\u09bc\u09c7\u09b0 \u0985\u09a8\u09c1\u09aa\u09be\u09a4\u09c7\u09b0 \u09b8\u09ae\u09be\u09a8\u0964<\/b><\/p>\n<p><span style=\"font-weight: 400;\">\u0985\u09b0\u09cd\u09a5\u09be\u09ce,\u00a0 BD : CD = AB : AC = c : b<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u098f\u0996\u09be\u09a8\u09c7, <\/span><span style=\"font-weight: 400;\">\u2206<\/span><span style=\"font-weight: 400;\">ABC \u098f\u09b0 \u09b6\u09c0\u09b0\u09cd\u09b7\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09b0 \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995<\/span><span style=\"font-weight: 400;\"> <span class=\"katex-eq\" data-katex-display=\"false\"> A\\left(x_{1}, y_{1}\\right), B\\left(x_{2}, y_{2}\\right) \\text{ \u098f\u09ac\u0982 }C\\left(x_{3}, y_{3}\\right) <\/span> \u09b9\u09b2\u09c7<\/span><\/p>\n<p><img loading=\"lazy\" class=\"aligncenter wp-image-3659 size-full\" src=\"https:\/\/10minuteschool.com\/content\/wp-content\/uploads\/2021\/12\/\u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f-\u09b0\u09c7\u0996\u09be\u09b0-\u09b8\u09be\u09aa\u09c7\u0995\u09cd\u09b7\u09c7-\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1-\u0993-\u09b0\u09c7\u0996\u09be\u0982\u09b6\u09c7\u09b0-\u09aa\u09cd\u09b0\u09a4\u09bf\u099a\u09cd\u099b\u09ac\u09bf.png\" alt=\"\u09b8\u09be\u09aa\u09c7\u0995\u09cd\u09b7\u09c7\" width=\"1052\" height=\"841\" srcset=\"https:\/\/10minuteschool.com\/content\/wp-content\/uploads\/2021\/12\/\u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f-\u09b0\u09c7\u0996\u09be\u09b0-\u09b8\u09be\u09aa\u09c7\u0995\u09cd\u09b7\u09c7-\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1-\u0993-\u09b0\u09c7\u0996\u09be\u0982\u09b6\u09c7\u09b0-\u09aa\u09cd\u09b0\u09a4\u09bf\u099a\u09cd\u099b\u09ac\u09bf.png 1052w, https:\/\/10minuteschool.com\/content\/wp-content\/uploads\/2021\/12\/\u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f-\u09b0\u09c7\u0996\u09be\u09b0-\u09b8\u09be\u09aa\u09c7\u0995\u09cd\u09b7\u09c7-\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1-\u0993-\u09b0\u09c7\u0996\u09be\u0982\u09b6\u09c7\u09b0-\u09aa\u09cd\u09b0\u09a4\u09bf\u099a\u09cd\u099b\u09ac\u09bf-300x240.png 300w, https:\/\/10minuteschool.com\/content\/wp-content\/uploads\/2021\/12\/\u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f-\u09b0\u09c7\u0996\u09be\u09b0-\u09b8\u09be\u09aa\u09c7\u0995\u09cd\u09b7\u09c7-\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1-\u0993-\u09b0\u09c7\u0996\u09be\u0982\u09b6\u09c7\u09b0-\u09aa\u09cd\u09b0\u09a4\u09bf\u099a\u09cd\u099b\u09ac\u09bf-1024x819.png 1024w, https:\/\/10minuteschool.com\/content\/wp-content\/uploads\/2021\/12\/\u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f-\u09b0\u09c7\u0996\u09be\u09b0-\u09b8\u09be\u09aa\u09c7\u0995\u09cd\u09b7\u09c7-\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1-\u0993-\u09b0\u09c7\u0996\u09be\u0982\u09b6\u09c7\u09b0-\u09aa\u09cd\u09b0\u09a4\u09bf\u099a\u09cd\u099b\u09ac\u09bf-768x614.png 768w\" sizes=\"(max-width: 1052px) 100vw, 1052px\" \/><\/p>\n<p><span style=\"font-weight: 400;\">D \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09b0 \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995, <span class=\"katex-eq\" data-katex-display=\"false\"> \\left(\\frac{c x_{3}+b x_{2}}{c+d}, \\frac{c y_{3}+b y_{2}}{c+d}\\right) <\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u0986\u09ac\u09be\u09b0, <span class=\"katex-eq\" data-katex-display=\"false\"> \\triangle \\mathrm{ABC} <\/span><\/span><span style=\"font-weight: 400;\">-\u098f <span class=\"katex-eq\" data-katex-display=\"false\"> \\mathrm{BI} \\angle \\mathrm{ABD} <\/span><\/span><span style=\"font-weight: 400;\">\u00a0\u098f\u09b0 \u09b8\u09ae\u09a6\u09cd\u09ac\u09bf\u0996\u09a3\u09cd\u09a1\u0995\u0964<\/span><\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\"> \\therefore \\frac{A I}{I D}=\\frac{A B}{B D} \\ldots \\quad \\cdots \\quad \\cdots \\text { (i) } <\/span>\n<p><span style=\"font-weight: 400;\">\u0986\u09ac\u09be\u09b0, <span class=\"katex-eq\" data-katex-display=\"false\"> \\triangle \\mathrm{ACD} <\/span><\/span><span style=\"font-weight: 400;\">-\u098f <span class=\"katex-eq\" data-katex-display=\"false\"> \\mathrm{BI} \\angle \\mathrm{ACD} <\/span><\/span><span style=\"font-weight: 400;\">\u00a0\u098f\u09b0 \u09b8\u09ae\u09a6\u09cd\u09ac\u09bf\u0996\u09a3\u09cd\u09a1\u0995\u0964<\/span><\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\"> \\therefore \\frac{A I}{I D}=\\frac{A C}{C D} \\ldots \\quad \\ldots \\quad \\ldots \\quad \\text { (ii) } <\/span>\n<p><span style=\"font-weight: 400;\">(i) \u0993 (ii) \u09a8\u0982 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u09b9\u09a4\u09c7 \u09aa\u09be\u0987,<\/span><\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\"> \\therefore \\frac{A I}{I D}=\\frac{A B}{B D}=\\frac{A C}{C D}=\\frac{A B+A C}{B D+C D} <\/span>\n<span class=\"katex-eq\" data-katex-display=\"false\"> \\therefore \\frac{A I}{I D}=\\frac{A B+A C}{B C} <\/span>\n<span class=\"katex-eq\" data-katex-display=\"false\"> \\therefore A I: I D=(c+b): a <\/span>\n<p><span style=\"font-weight: 400;\">\u2234\u00a0<\/span><span style=\"font-weight: 400;\">I \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09b0 \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995, <span class=\"katex-eq\" data-katex-display=\"false\"> \\left(\\frac{(c+b) \\frac{c x_{3}+b x_{2}}{(c+b)}+a \\cdot x_{1}}{c+b+a}, \\frac{(c+b) \\frac{c y_{3}+b y_{2}}{(c+b)} a \\cdot y_{1}}{c+b+a}\\right) <\/span><\/span><\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\"> =\\left(\\frac{a x_{1}+b x_{2}+c x_{3}}{a+b+c}, \\frac{a y_{1}+b y_{2}+c y_{3}}{a+b+c}\\right) <\/span>\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\therefore(x_{1}, y_{2}),(x_{2}, y_{2}) \\text { \u098f\u09ac\u0982 }(x_{3}, y_{3}) <\/span> \u09b6\u09c0\u09b0\u09cd\u09b7\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09cd\u09b0\u09af\u09bc \u09a6\u09c7\u09af\u09bc\u09be \u09a5\u09be\u0995\u09b2\u09c7 \u09a4\u09cd\u09b0\u09bf\u09ad\u09c1\u099c\u09c7\u09b0 \u0985\u09a8\u09cd\u09a4\u0983\u0995\u09c7\u09a8\u09cd\u09a6\u09cd\u09b0<\/span> <span class=\"katex-eq\" data-katex-display=\"false\"> \\left(\\frac{a x_{1}+b x_{2}+c x_{3}}{a+b+c}, \\frac{a y_{1}+b y_{2}+c y_{3}}{a+b+c}\\right) <\/span><\/p>\n<div class=\"x1tlxs6b x1g8br2z x1gn5b1j x230xth x14ctfv x1okitfd x6ikm8r x10wlt62 x1mzt3pk x1y1aw1k xn6708d xwib8y2 x1ye3gou x1n2onr6 x13faqbe x1vjfegm\" role=\"none\">\n<div class=\"\">\n<div class=\"x9f619 x1n2onr6 x1ja2u2z __fb-light-mode\" role=\"none\">\n<hr \/>\n<p dir=\"auto\" role=\"none\">\n<p class=\"x6prxxf x1fc57z9 x1yc453h x126k92a xzsf02u\" dir=\"auto\" role=\"none\"><em><strong>\u098f\u0987\u099a\u098f\u09b8\u09b8\u09bf \u0993 \u098f\u09a1\u09ae\u09bf\u09b6\u09a8 \u09aa\u09b0\u09c0\u0995\u09cd\u09b7\u09be\u09b0\u09cd\u09a5\u09c0\u09a6\u09c7\u09b0 \u099c\u09a8\u09cd\u09af \u0986\u09ae\u09be\u09a6\u09c7\u09b0 \u0995\u09cb\u09b0\u09cd\u09b8\u09b8\u09ae\u09c2\u09b9\u0983<\/strong><\/em><\/p>\n<\/div>\n<\/div>\n<\/div>\n<ul>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/hsc-25-online-batch-2-bangla-english-ict\/\">HSC 25 \u0985\u09a8\u09b2\u09be\u0987\u09a8 \u09ac\u09cd\u09af\u09be\u099a \u09e8.\u09e6 (\u09ac\u09be\u0982\u09b2\u09be, \u0987\u0982\u09b0\u09c7\u099c\u09bf, \u09a4\u09a5\u09cd\u09af \u0993 \u09af\u09cb\u0997\u09be\u09af\u09cb\u0997 \u09aa\u09cd\u09b0\u09af\u09c1\u0995\u09cd\u09a4\u09bf)<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/hsc-26-online-batch-bangla-english-ict\/\">HSC 26 \u0985\u09a8\u09b2\u09be\u0987\u09a8 \u09ac\u09cd\u09af\u09be\u099a (\u09ac\u09be\u0982\u09b2\u09be, \u0987\u0982\u09b0\u09c7\u099c\u09bf, \u09a4\u09a5\u09cd\u09af \u0993 \u09af\u09cb\u0997\u09be\u09af\u09cb\u0997 \u09aa\u09cd\u09b0\u09af\u09c1\u0995\u09cd\u09a4\u09bf)<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/hsc-2025-online-batch\/\">HSC 25 \u0985\u09a8\u09b2\u09be\u0987\u09a8 \u09ac\u09cd\u09af\u09be\u099a (\u09ab\u09bf\u099c\u09bf\u0995\u09cd\u09b8, \u0995\u09c7\u09ae\u09bf\u09b8\u09cd\u099f\u09cd\u09b0\u09bf, \u09ae\u09cd\u09af\u09be\u09a5, \u09ac\u09be\u09df\u09cb\u09b2\u099c\u09bf)<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/hsc-2026-online-batch\/\">HSC 26 \u0985\u09a8\u09b2\u09be\u0987\u09a8 \u09ac\u09cd\u09af\u09be\u099a (\u09ab\u09bf\u099c\u09bf\u0995\u09cd\u09b8, \u0995\u09c7\u09ae\u09bf\u09b8\u09cd\u099f\u09cd\u09b0\u09bf, \u09ae\u09cd\u09af\u09be\u09a5, \u09ac\u09be\u09df\u09cb\u09b2\u099c\u09bf)<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/medical-admission-course\/\">\u09ae\u09c7\u09a1\u09bf\u0995\u09c7\u09b2 \u098f\u09a1\u09ae\u09bf\u09b6\u09a8 \u0995\u09cb\u09b0\u09cd\u09b8 - \u09e8\u09e6\u09e8\u09ea<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/dhaka-university-a-unit-admission-course\/\">\u09a2\u09be\u0995\u09be \u09ad\u09be\u09b0\u09cd\u09b8\u09bf\u099f\u09bf A Unit \u098f\u09a1\u09ae\u09bf\u09b6\u09a8 \u0995\u09cb\u09b0\u09cd\u09b8 - \u09e8\u09e6\u09e8\u09ea<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/dhaka-university-b-unit-admission-course\/\">\u09a2\u09be\u0995\u09be \u09ad\u09be\u09b0\u09cd\u09b8\u09bf\u099f\u09bf B Unit \u098f\u09a1\u09ae\u09bf\u09b6\u09a8 \u0995\u09cb\u09b0\u09cd\u09b8 - \u09e8\u09e6\u09e8\u09ea<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/buet-ques-solve\/\">\u09ac\u09c1\u09df\u09c7\u099f \u0995\u09cb\u09b6\u09cd\u099a\u09c7\u09a8 \u09b8\u09b2\u09ad \u0995\u09cb\u09b0\u09cd\u09b8<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/gst-a-unit-admission-course\/\">\u0997\u09c1\u099a\u09cd\u099b A Unit \u098f\u09a1\u09ae\u09bf\u09b6\u09a8 \u0995\u09cb\u09b0\u09cd\u09b8 - \u09e8\u09e6\u09e8\u09ea<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/gst-b-unit-admission-course\/\">\u0997\u09c1\u099a\u09cd\u099b B Unit \u098f\u09a1\u09ae\u09bf\u09b6\u09a8 \u0995\u09cb\u09b0\u09cd\u09b8 - \u09e8\u09e6\u09e8\u09ea<\/a><\/span><\/li>\n<\/ul>\n<hr \/>\n<p>&nbsp;<\/p>\n<p><em><strong>\u0986\u09ae\u09be\u09a6\u09c7\u09b0 \u09b8\u09cd\u0995\u09bf\u09b2 \u09a1\u09c7\u09ad\u09c7\u09b2\u09aa\u09ae\u09c7\u09a8\u09cd\u099f \u0995\u09cb\u09b0\u09cd\u09b8\u09b8\u09ae\u09c2\u09b9\u0983<\/strong><\/em><\/p>\n<ul>\n<li><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/study-abroad-complete-guideline\/\">\u09ac\u09bf\u09a6\u09c7\u09b6\u09c7 \u0989\u099a\u09cd\u099a\u09b6\u09bf\u0995\u09cd\u09b7\u09be: Study Abroad Complete Guideline<\/a><\/span><\/li>\n<li><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/student-hacks\/\">Student Hacks<\/a><\/span><\/li>\n<li><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/ielts-course\/\">IELTS Course by Munzereen Shahid<\/a><\/span><\/li>\n<li><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/english-grammar-course\/\">Complete English Grammar Course<\/a><\/span><\/li>\n<li><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/ms-bundle\/\"> Microsoft Office 3 in 1 Bundle<\/a><\/span><\/li>\n<li><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/ghore-boshe-freelancing\/\">\u0998\u09b0\u09c7 \u09ac\u09b8\u09c7 Freelancing<\/a><\/span><\/li>\n<li><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/facebook-marketing\/\">Facebook Marketing<\/a><\/span><\/li>\n<li><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/adobe-4-in-1-bundle\/\">Adobe 4 in 1 Bundle<\/a><\/span><\/li>\n<\/ul>\n<hr \/>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\"><em>\u09e7<\/em><em>\u09e6 \u09ae\u09bf\u09a8\u09bf\u099f \u09b8\u09cd\u0995\u09c1\u09b2\u09c7\u09b0 \u0995\u09cd\u09b2\u09be\u09b8\u0997\u09c1\u09b2\u09cb \u0985\u09a8\u09c1\u09b8\u09b0\u09a3 \u0995\u09b0\u09a4\u09c7 \u09ad\u09bf\u099c\u09bf\u099f: <span style=\"color: #993300;\"><strong><a style=\"color: #993300;\" href=\"https:\/\/10minuteschool.com\/?ref=https%3A%2F%2Fblog.10minuteschool.com%2Fwordpress%2F&amp;post_id=78178&amp;blog_category_id=700\">www.10minuteschool.com<\/a><\/strong><\/span><\/em><\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09be\u09aa\u09c7\u0995\u09cd\u09b7\u09c7 \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u0993 \u09b0\u09c7\u0996\u09be\u0982\u09b6\u09c7\u09b0 \u09aa\u09cd\u09b0\u09a4\u09bf\u099a\u09cd\u099b\u09ac\u09bf (Image of a point and a straight line with respect to a definite straight line) \u0986\u09ae\u09bf \u09af\u09a6\u09bf \u0986\u09af\u09bc\u09a8\u09be\u09b0 \u09b8\u09be\u09ae\u09a8\u09c7 \u098f\u0995\u099f\u09bf \u0995\u09b2\u09ae\u0995\u09c7 \u09a7\u09b0\u09bf \u09b8\u09c7\u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7 \u09b2\u0995\u09cd\u09b7\u09cd\u09af \u0995\u09b0\u09be \u09af\u09be\u09af\u09bc \u09af\u09c7, \u0986\u09af\u09bc\u09a8\u09be\u09b0 \u09aa\u09c3\u09b7\u09cd\u09a0 \u09a5\u09c7\u0995\u09c7 \u09a0\u09bf\u0995 \u09af\u09a4 \u09a6\u09c2\u09b0\u09c7<\/p>\n<p> <a class=\"redmore\" href=\"https:\/\/10minuteschool.com\/content\/definite-straight-line\/\">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":[2529,2528,2527,2530,490],"_links":{"self":[{"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/posts\/3648"}],"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=3648"}],"version-history":[{"count":11,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/posts\/3648\/revisions"}],"predecessor-version":[{"id":16098,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/posts\/3648\/revisions\/16098"}],"wp:attachment":[{"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/media?parent=3648"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/categories?post=3648"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/tags?post=3648"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}