{"id":3664,"date":"2024-01-30T12:20:56","date_gmt":"2024-01-30T06:20:56","guid":{"rendered":"https:\/\/stage-wp.10minuteschool.com\/?p=3664"},"modified":"2024-10-29T17:48:11","modified_gmt":"2024-10-29T11:48:11","slug":"equation-of-straight-line","status":"publish","type":"post","link":"https:\/\/10minuteschool.com\/content\/equation-of-straight-line\/","title":{"rendered":"\u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 | Equation of Straight Line"},"content":{"rendered":"

\u09ac\u09bf\u09ad\u09bf\u09a8\u09cd\u09a8 \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u09b8\u09c1\u09a4\u09cd\u09b0 \u09b8\u09ae\u09c2\u09b9<\/span><\/h2>\n

\u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3<\/strong> \u09b9\u09b2 \u098f\u0995\u099f\u09bf \u0997\u09be\u09a3\u09bf\u09a4\u09bf\u0995 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u09af\u09be \u098f\u0995\u099f\u09bf \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u0989\u09aa\u09b0 \u0985\u09ac\u09b8\u09cd\u09a5\u09bf\u09a4 \u09b8\u0995\u09b2 \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09b0 x \u098f\u09ac\u0982 y \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995\u09c7\u09b0 \u09ae\u09a7\u09cd\u09af\u0995\u09be\u09b0 \u09b8\u09ae\u09cd\u09aa\u09b0\u09cd\u0995\u0995\u09c7 \u09aa\u09cd\u09b0\u0995\u09be\u09b6 \u0995\u09b0\u09c7\u0964 \u098f\u0987 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u099f\u09bf \u09b8\u09be\u09a7\u09be\u09b0\u09a3\u09a4 x \u098f\u09ac\u0982 y \u099a\u09b2\u0995\u09c7\u09b0 \u098f\u0995\u099f\u09bf \u098f\u0995\u0998\u09be\u09a4 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09c7\u09b0 \u0986\u0995\u09be\u09b0\u09c7 \u09a5\u09be\u0995\u09c7\u0964 \u09b8\u09ac\u099a\u09c7\u09af\u09bc\u09c7 \u09b8\u09be\u09a7\u09be\u09b0\u09a3 \u0986\u0995\u09be\u09b0\u09c7\u09b0\u00a0\u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u09b8\u09c2\u09a4\u09cd\u09b0 \u09b9\u09b2:<\/p>\n

\"\u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0<\/p>\n

(\u0995) x<\/span> \u0985\u0995\u09cd\u09b7\u09c7\u09b0 \u09b8\u09ae\u09be\u09a8\u09cd\u09a4\u09b0\u09be\u09b2 \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be<\/a><\/span> AB \u098f\u09b0 \u09b8\u0995\u09b2 \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09b0 \u0995\u09cb\u099f\u09bf y = b<\/span><\/strong><\/p>\n

\\therefore x<\/span><\/span> \u0985\u0995\u09cd\u09b7\u09c7\u09b0 \u09b8\u09ae\u09a8\u09cd\u09a4\u09b0\u09b2 \u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 y = b<\/span><\/span><\/p>\n

\"\u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0<\/p>\n

\u09af\u09c7\u0996\u09be\u09a8\u09c7 b \u09b9\u09b2 x \u0985\u0995\u09cd\u09b7 \u09b9\u09a4\u09c7 AB \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u09a6\u09c2\u09b0\u09a4\u09cd\u09ac\u0964\u00a0<\/span><\/p>\n

\u0985\u09a8\u09c1\u09b8\u09bf\u09a6\u09cd\u09a7\u09be\u09a8\u09cd\u09a4 (\u09e7)\u00a0<\/b><\/p>\n

x<\/span> \u0985\u0995\u09cd\u09b7\u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3<\/a><\/span> y = 0<\/span><\/b><\/p>\n

\u0985\u09a8\u09c1\u09b8\u09bf\u09a6\u09cd\u09a7\u09be\u09a8\u09cd\u09a4 (\u09e8)<\/b><\/p>\n

x<\/span>–<\/span> \u0985\u0995\u09cd\u09b7\u09c7\u09b0 \u09b8\u09ae\u09be\u09a8\u09cd\u09a4\u09b0\u09b2 \u09b0\u09c7\u0996\u09be\u0995\u09c7 <\/span>y<\/span>–<\/span> \u0985\u0995\u09cd\u09b7\u09c7\u09b0 \u0989\u09aa\u09b0 \u09b2\u09ae\u09cd\u09ac \u09b0\u09c7\u0996\u09be\u0993 \u09ac\u09b2\u09be \u09b9\u09df\u0964 \u09b8\u09c1\u09a4\u09b0\u09be\u0982, y<\/span><\/span>–<\/span>\u00a0<\/span><\/p>\n

\u0985\u0995\u09cd\u09b7\u09c7\u09b0 \u0989\u09aa\u09b0 \u09b2\u09ae\u09cd\u09ac \u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 <\/b>y = b<\/span><\/span>.\u00a0<\/b><\/p>\n

(\u0996) y<\/span>\u00a0\u0985\u0995\u09cd\u09b7\u09c7\u09b0 \u09b8\u09ae\u09be\u09a8\u09cd\u09a4\u09b0\u09be\u09b2 \u09b0\u09c7\u0996\u09be A B<\/span>– \u098f\u09b0 \u09b8\u0995\u09b2 \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09b0 \u09ad\u09c1\u099c x = a<\/span><\/strong><\/h4>\n

\\therefore y<\/span><\/span> \u0985\u0995\u09cd\u09b7\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 <\/b>x = a<\/span><\/span>\u00a0<\/b><\/p>\n

\"\u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0<\/p>\n

\u09af\u09c7\u0996\u09be\u09a8\u09c7 <\/span>a<\/span><\/span> \u09b9\u09b2\u09c7 <\/span>y<\/span><\/span> \u0985\u0995\u09cd\u09b7 \u09a5\u09c7\u0995\u09c7 <\/span>A B<\/span><\/span> \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u09a6\u09c2\u09b0\u09a4\u09cd\u09ac\u0964\u00a0<\/span><\/p>\n

\u0985\u09a8\u09c1\u09b8\u09bf\u09a6\u09cd\u09a7\u09be\u09a8\u09cd\u09a4 (\u09e7)<\/b><\/p>\n

y<\/span> \u0985\u0995\u09cd\u09b7\u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u00a0 x = 0<\/span><\/b>\u00a0<\/b><\/p>\n

\u0985\u09a8\u09c1\u09b8\u09bf\u09a6\u09cd\u09a7\u09be\u09a8\u09cd\u09a4 (\u09e8)<\/b><\/p>\n

y<\/span>–<\/span> \u0985\u0995\u09cd\u09b7\u09c7\u09b0 \u09b8\u09ae\u09be\u09a8\u09cd\u09a4\u09b0\u09be\u09b2 \u09b0\u09c7\u0996\u09be\u0995\u09c7 <\/span>x<\/span>–<\/span> \u0985\u0995\u09cd\u09b7\u09c7\u09b0 \u0989\u09aa\u09b0 \u09b2\u09ae\u09cd\u09ac \u09b0\u09c7\u0996\u09be\u0993 \u09ac\u09b2\u09be \u09b9\u09df \u0964<\/span><\/p>\n

\u0985\u0995\u09cd\u09b7\u09c7\u09b0 \u0989\u09aa\u09b0 \u09b2\u09ae\u09cd\u09ac \u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 <\/span>x = a<\/span>.<\/span><\/p>\n

(\u0997) \u09ae\u09c2\u09b2 \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u0997\u09be\u09ae\u09c0 \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be :<\/b><\/p>\n

\"\u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0<\/p>\n

\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, \u09ae\u09c2\u09b2\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u0997\u09be\u09ae\u09c0 AB \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u0989\u09aa\u09b0\u09c7 P(x, y) \u09af\u09c7 \u0995\u09cb\u09a8 \u098f\u0995\u099f\u09bf \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u0964 P \u09b9\u09a4\u09c7 x \u0985\u0995\u09cd\u09b7\u09c7\u09b0 \u0989\u09aa\u09b0\u09c7 PL \u09b2\u09ae\u09cd\u09ac \u099f\u09be\u09a8\u09bf\u0964<\/span><\/p>\n

\u09a7\u09b0\u09bf, \\angle P O L = \\theta<\/span><\/span><\/p>\n

\\therefore \\triangle POL<\/span><\/span> \u09b9\u09a4\u09c7 \u09aa\u09be\u0987, \\tan \\theta=\\frac{P L}{O L}=\\frac{y}{x} \\Rightarrow(\\tan \\theta) x=m x=y<\/span><\/span><\/p>\n

\u09af\u09c7\u0996\u09be\u09a8\u09c7 <\/span>\\tan \\theta=m<\/span><\/span> \u0995\u09c7 <\/span>A B<\/span><\/span> \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u09a2\u09be\u09b2 \u09ac\u09be \u0995\u09cd\u09b0\u09ae\u09be\u09ac\u09a8\u09a4\u09bf \u09ac\u09b2\u09be \u09b9\u09df\u0964 \u09b8\u09c1\u09a4\u09b0\u09be\u0982 <\/span>\u09ae\u09c2\u09b2\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>y=m x<\/span><\/span>\u00a0<\/b><\/p>\n

(\u0995) y<\/span> \u0985\u0995\u09cd\u09b7 \u09b9\u09a4\u09c7 c<\/span> \u0985\u0982\u09b6 \u099b\u09bf\u09a8\u09cd\u09a8\u0995\u09be\u09b0\u09c0 \u098f\u09ac\u0982 \u09a7\u09a8\u09be\u09a4\u09cd\u09ac\u0995 x<\/span> \u0985\u0995\u09cd\u09b7\u09c7\u09b0 \u09b8\u0982\u0997\u09c7 \\theta<\/span>\u00a0 \u0995\u09cb\u09a8\u09c7 \u0986\u09a8\u09a4\u0964<\/strong><\/p>\n

\"\u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0<\/p>\n

\u09a7\u09b0\u09bf, \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u0989\u09aa\u09b0 <\/b>P(x, y)<\/span>\u00a0\u09af\u09c7 \u0995\u09cb\u09a8 \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u0964\u00a0\u00a0<\/b><\/p>\n

x<\/span> \u0985\u0995\u09cd\u09b7\u09b0\u09c7\u0996\u09be\u09b0 \u0989\u09aa\u09b0\u09c7 <\/span>P L<\/span><\/span> \u09b2\u09ae\u09cd\u09ac \u099f\u09be\u09a8\u09bf \u098f\u09ac\u0982 <\/span>C<\/span><\/span> \u09b9\u09a4\u09c7 <\/span>P L<\/span><\/span> \u098f\u09b0 \u0989\u09aa\u09b0 <\/span>C M<\/span><\/span> \u09b2\u09ae\u09cd\u09ac \u099f\u09be\u09a8\u09bf\u0964\u00a0\u00a0<\/span><\/p>\n

\u098f\u0996\u09be\u09a8\u09c7, <\/span>O C=c, \\angle A B O=\\angle A B X=\\theta<\/span><\/span><\/p>\n\\therefore C M=O L=x, P M=P L-M L=P L-C O=y-c<\/span>\n

\u098f\u09ac\u0982 <\/span>\\angle P C M=\\angle A B X=\\theta<\/span><\/span>\u00a0<\/span><\/p>\n\\therefore \\tan \\theta=\\frac{P M}{C M}=\\frac{y-c}{x}<\/span>\n\\Rightarrow m=\\frac{y-c}{x}[\\therefore \\tan \\theta=m]<\/span>\n\\Rightarrow y-c=m x<\/span>\n

\\Rightarrow y=m x+c<\/span>,<\/span> \u0987\u09b9\u09be <\/b>A B<\/span><\/span> \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u0964<\/b><\/p>\n

\u0985\u09a8\u09c1\u09b8\u09bf\u09a6\u09cd\u09a7\u09be\u09a8\u09cd\u09a4 (\u09e7)<\/b><\/p>\n

a x+b y+c=0<\/span><\/span> \u09b0\u09c7\u0996\u09be\u09b0 \u09a2\u09be\u09b2 =\\frac{-x \\text { \u098f\u09b0 \u09b8\u09b9\u0997 }}{y \\text { \u098f\u09b0 \u09b8\u09b9\u0997 }}<\/span><\/span> \u0985\u09b0\u09cd\u09a5\u09be\u09ce, m=-\\frac{a}{b}<\/span><\/span>\u00a0\u0964<\/span><\/p>\n

\u0995\u09be\u09b0\u09a3 b y=-c-a x \\Rightarrow y=\\frac{-a}{b} \\times-\\frac{c}{b}<\/span><\/span><\/p>\n

(2) <\/span>C<\/span><\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09b0 \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995 =(0, c)<\/span><\/span><\/p>\n

(\u0999) (x, y) \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u0997\u09be\u09ae\u09c0 \u098f\u09ac\u0982 m \u09a2\u09be\u09b2\u09ac\u09bf\u09b6\u09bf\u09b7\u09cd\u099f \u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3,<\/strong><\/p>\n

\"\u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0<\/p>\n

\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, \\mathrm{AB}<\/span> \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u099f\u09bf \\mathrm{Q}\\left(x_{1} y_{1}\\right)<\/span><\/span>\u00a0\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09a6\u09bf\u09af\u09bc\u09c7 \u09af\u09be\u09af\u09bc \u098f\u09ac\u0982 \u09b0\u09c7\u0996\u09be\u099f\u09bf\u09b0 \u09a2\u09be\u09b2 m. \u09a7\u09b0\u09bf, AB \u09b0\u09c7\u0996\u09be\u09b0 \u0989\u09aa\u09b0 <\/span>P(x, y)<\/span> \u09af\u09c7\u0995\u09cb\u09a8\u09cb \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u0964<\/span><\/p>\n

\u09a4\u09be\u09b9\u09b2\u09c7, PQ \u098f\u09b0 \u09a2\u09be\u09b2 =\\frac{y-y_{1}}{x-x_{1}}[: \\mathrm{AB}<\/span><\/span> = \\mathrm{PQ}<\/span> \u09b0\u09c7\u0996\u09be\u09b0 \u09a2\u09be\u09b2]<\/span><\/p>\n\\therefore y-y_{1}=m\\left(x-x_{1}\\right)<\/span>\n

 <\/p>\n

(\u099a) \u09a6\u09c1\u0987\u099f\u09bf \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09a6\u09bf\u09af\u09bc\u09c7 \u0985\u09a4\u09bf\u0995\u09cd\u09b0\u09ae\u0995\u09be\u09b0\u09c0 \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3:\u00a0<\/b><\/p>\n

\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, AB \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u099f\u09bf Q\\left(x_{1}, y_{1}\\right)<\/span><\/span> \u0993 R\\left(x_{2}, y_{2}\\right)<\/span><\/span> \u09a6\u09c1\u0987\u099f\u09bf \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09a6\u09bf\u09af\u09bc\u09c7 \u0985\u09a4\u09bf\u0995\u09cd\u09b0\u09ae \u0995\u09b0\u09c7 \u098f\u09ac\u0982 \u09b0\u09c7\u0996\u09be\u099f\u09bf\u09b0 \u0989\u09aa\u09b0 P(x, y)<\/span> \u09af\u09c7 \u0995\u09cb\u09a8\u09cb \u098f\u0995\u099f\u09bf \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1<\/span>\u00a0<\/b><\/p>\n

P Q<\/span> <\/span>\u098f\u09b0<\/span>\u00a0<\/span>\u09a2\u09be\u09b2 =\\frac{y-y_{1}}{x-x_{1}}<\/span><\/span><\/p>\n

Q R<\/span> <\/span>\u098f\u09b0<\/span>\u00a0<\/span>\u09a2\u09be\u09b2 =\\frac{y_{1}-y_{2}}{x_{1}-x_{2}}<\/span><\/span>\u00a0\u00a0<\/b><\/p>\n

P . Q . R<\/span>\u00a0<\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u0995\u09cd\u09b0\u09df \u09b8\u09ae\u09b0\u09c7\u0996 \u09ac\u09b2\u09c7,<\/b>\u00a0\u00a0<\/b><\/p>\n

\"\u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0<\/p>\n

P Q<\/span><\/span> \u098f\u09b0 \u09a2\u09be\u09b2 <\/span>= Q R<\/span><\/span> \u098f\u09b0 \u09a2\u09be\u09b2\u00a0\u00a0\u00a0<\/span><\/p>\n\\frac{y-y_{1}}{x-x_{1}}=\\frac{y_{1}-y_{2}}{x_{1}-x_{2}}<\/span>\n

\u09ac\u09be, \\frac{y-y_{1}}{y_{1}-y_{2}}=\\frac{x-x_{1}}{x_{1}-x_{2}}<\/span><\/span>\u00a0\u00a0<\/span><\/p>\n\\therefore y-y_{1}=\\frac{y_{1}-y_{2}}{x_{1}-x_{2}}\\left(x-x_{1}\\right)<\/span>\n

\u0989\u09b2\u09cd\u09b2\u09c7\u0996\u09cd\u09af , \u098f\u0996\u09be\u09a8\u09c7 \\frac{y_{1}-y_{2}}{x_{1}-x_{2}}=m=<\/span> <\/span>\u09b0\u09c7\u0996\u09be\u099f\u09bf\u09b0<\/span>\u00a0<\/span>\u09a2\u09be\u09b2<\/span><\/p>\n

\u0985\u09a8\u09c1\u09b8\u09bf\u09a6\u09cd\u09a7\u09be\u09a8\u09cd\u09a4 : (\u09e7)\u00a0\u00a0<\/b><\/p>\n

\u09ae\u09c2\u09b2\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 (0,0)<\/span> \u098f\u09ac\u0982 \\left(x_{1}, y_{1}\\right)<\/span><\/span>\u00a0\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<\/span><\/p>\ny=\\frac{Y_{1}}{X_{1}} x<\/span>\n

 <\/p>\n

(\u099b) \u099b\u09c7\u09a6\u0995 \u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u00a0\u00a0\u00a0<\/b><\/p>\n

\"\u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0<\/p>\n

\u09a7\u09b0\u09bf, AB \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u099f\u09bf x \u0993 y \u0985\u0995\u09cd\u09b7 \u09b9\u09a4\u09c7 \u09af\u09a5\u09be\u0995\u09cd\u09b0\u09ae\u09c7 a \u0993 b \u0985\u0982\u09b6 \u099b\u09bf\u09a8\u09cd\u09a8 \u0995\u09b0\u09c7\u099b\u09c7 \u098f\u09ac\u0982 AB \u098f\u09b0 \u0989\u09aa\u09b0\u00a0 <\/span>P(x, y)<\/span> \u09af\u09c7 \u0995\u09cb\u09a8 \u098f\u0995\u099f\u09bf \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u0964 x<\/span> \u0993 y<\/span> \u0985\u0995\u09cd\u09b7\u09c7\u09b0 \u0989\u09aa\u09b0\u09c7 \u09af\u09a5\u09be\u0995\u09cd\u09b0\u09ae\u09c7 PL \u0993 PM \u09b2\u09ae\u09cd\u09ac \u099f\u09be\u09a8\u09bf \u098f\u09ac\u0982 OP \u09af\u09c1\u0995\u09cd\u09a4 \u0995\u09b0\u09bf\u0964<\/span><\/p>\n

\u098f\u0996\u09be\u09a8\u09c7, O A=a, O B=b ; O L=P M=x, P L=y<\/span><\/span><\/p>\n\\therefore \\Delta O A B=\\Delta O P A+\\Delta O P B<\/span>\n

\\Rightarrow \\frac{1}{2} O A \\cdot O B=\\frac{1}{2} O A \\cdot P L+\\frac{1}{2} O B \\cdot P M<\/span>\u00a0<\/span><\/p>\n

\\Rightarrow \\frac{1}{2} a b=\\frac{1}{2} a \\cdot y+\\frac{1}{2} b \\cdot x<\/span>\u00a0<\/span><\/p>\n

\\Rightarrow a b=a y+b x<\/span><\/span>\u00a0<\/span><\/p>\n

\\Rightarrow 1=\\frac{b x+a y}{a b}<\/span>\u00a0<\/span><\/p>\n

\\Rightarrow \\frac{x}{a}+\\frac{y}{b}=1<\/span> \u0987\u09b9\u09be \\mathrm{AB}<\/span> \u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u00a0<\/span><\/p>\n

\u0985\u09a8\u09c1\u09b8\u09bf\u09a6\u09cd\u09a7\u09be\u09a8\u09cd\u09a4 (\u09e7)<\/b><\/p>\n

\\frac{x}{a}+\\frac{y}{b}=1<\/span> \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be <\/span>x<\/span> <\/span>\u0993<\/span> y<\/span><\/span> \u0985\u0995\u09cd\u09b7\u0995\u09c7 \u09af\u09a5\u09be\u0995\u09cd\u09b0\u09ae\u09c7 <\/span>A(a, 0)<\/span><\/span> \u098f\u09ac\u0982 <\/span>B(0, b)<\/span><\/span> \u09ac\u09bf\u09ac\u09cd\u09a6\u09c1\u09a4\u09c7 \u099b\u09c7\u09a6 \u0995\u09b0\u09c7\u0964<\/span><\/p>\n

 <\/p>\n

(\u099c) \u09a6\u09c1\u0987\u099f\u09bf \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u099b\u09c7\u09a6\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09a8\u09bf\u09b0\u09cd\u09a3\u09df\u09c7\u09b0 \u09aa\u09a6\u09cd\u09a7\u09cd\u09ac\u09a4\u09bf :\u00a0<\/b><\/p>\n\\begin{array}{l}\n\na_{1} x+b_{1} y+c_{1}=0-(i) \\\\\n\na_{2} x+b_{2} y+c_{2}=0-(i i)\n\n\\end{array}<\/span>\n

(i)<\/span> \u0993<\/span>\u00a0(ii)<\/span> \u09a8\u0982 \u09b9\u09a4\u09c7 \u09ac\u099c\u09cd\u09b0\u0997\u09c1\u09a8\u09c7\u09b0 \u09b8\u09c1\u09a4\u09cd\u09b0 \u0985\u09a8\u09c1\u09b8\u09be\u09b0\u09c7 \u09aa\u09be\u0987,\u00a0\u00a0<\/span><\/p>\n\\begin{array}{l}\n\n\\frac{x}{b_{1} c_{2}-b_{2} c_{1}}=\\frac{y}{a_{2} c_{1}-a_{1} c_{2}}=\\frac{1}{a_{1} b_{2}-a_{2} b_{1}} \\\\\n\n\\Rightarrow x=\\frac{b_{1} c_{2}-b_{2} c_{1}}{a_{1} b_{2}-a_{2} b_{1}}, y=\\frac{a_{2} c_{1}-a_{1} c_{2}}{a_{1} b_{2}-a_{2} b_{1}}\n\n\\end{array}<\/span>\n

\u09af\u09c7\u09ae\u09a8, <\/span>3 x-4 y+6=0 ; 4 x+3 y-7=0<\/span><\/span>\u00a0\u00a0<\/span><\/p>\n

\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, \\frac{x}{28-18}=\\frac{y}{24+21}=\\frac{1}{9+16}<\/span><\/span><\/p>\n\\therefore x=\\frac{10}{25}=\\frac{2}{5} ; y=\\frac{45}{25}=\\frac{9}{5}<\/span>\n

\u09b8\u09c1\u09a4\u09b0\u09be\u0982x=\\frac{b_{1} c_{2}-b_{2} c_{1}}{a_{1} b_{2}-a_{2} b_{1}}<\/span> \u098f\u09ac\u0982 y=\\frac{a_{2} c_{1}-a_{1} c_{2}}{a_{1} b_{2}-a_{2} b_{1}}<\/span><\/span><\/p>\n

\u09a6\u09bf\u09df\u09c7 \u09b8\u09b0\u09be\u09b8\u09b0\u09bf \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09a6\u09cd\u09ac\u09c7\u09df\u09c7\u09b0 \u099b\u09c7\u09a6\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09a8\u09bf\u09b0\u09cd\u09a3\u09df \u0995\u09b0\u09be \u09af\u09be\u09df\u0964\u00a0<\/span><\/p>\n

(\u099d) \u09ae\u09c2\u09b2\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09a5\u09c7\u0995\u09c7 \u0995\u09cb\u09a8 \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u0989\u09aa\u09b0 \u0985\u0999\u09cd\u0995\u09bf\u09a4 \u09b2\u09ae\u09cd\u09ac\u09c7\u09b0 \u09a6\u09c8\u09b0\u09cd\u0998\u09cd\u09af p<\/span> \u098f\u09ac\u0982 \u09b2\u09ae\u09cd\u09ac\u099f\u09bf x<\/span> \u0985\u0995\u09cd\u09b7\u09c7\u09b0 \u09a7\u09a8\u09be\u09a4\u09cd\u09ae\u0995 \u09a6\u09bf\u0995\u09c7\u09b0 \u09b8\u0982\u0999\u09cd\u0997\u09c7 <\/span> \\alpha<\/span> \u0995\u09cb\u09a3 \u0989\u09ce\u09aa\u09a8\u09cd\u09a8 \u0995\u09b0\u09b2\u09c7,\u00a0<\/span><\/p>\n

\u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3x \\cos \\alpha+y \\sin \\alpha=p<\/span><\/span>\u00a0\u00a0<\/span><\/p>\n

\"\u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0<\/p>\n

\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, \u09ae\u09c2\u09b2\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09a5\u09c7\u0995\u09c7 \\mathrm{AB}<\/span> \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u0989\u09aa\u09b0 \u09b2\u09ae\u09cd\u09ac <\/span>O Q<\/span><\/span> \u098f\u09ac\u0982 <\/span>\\angle Q O A=\\alpha<\/span><\/b><\/span>\u00a0<\/span><\/p>\n

\\Delta O A Q<\/span><\/span> \u09b9\u09a4\u09c7 \u09aa\u09be\u0987, \\frac{O Q}{O A}=\\cos \\alpha<\/span><\/span>\u00a0<\/span><\/p>\n

\u09ac\u09be, O A=\\frac{O Q}{\\cos \\alpha}=\\frac{p}{\\cos \\alpha}=p \\sec \\alpha<\/span><\/span>\u00a0<\/span><\/p>\n\\therefore O A=p \\sec \\alpha<\/span>\n

\\triangle O B Q<\/span><\/span> \u09b9\u09a4\u09c7 \u09aa\u09be\u0987, <\/span>\\angle B O Q=90^{\\circ}-\\alpha<\/span><\/b><\/span><\/p>\n\\therefore \\frac{O Q}{O B}=\\cos \\left(90^{\\circ}-\\alpha\\right)=\\sin \\alpha<\/span>\n

\u09ac\u09be, O B=\\frac{O Q}{\\sin \\alpha}=\\frac{p}{\\sin \\alpha}=p \\operatorname{cosec} \\alpha<\/span><\/span><\/p>\n

\u09b8\u09c1\u09a4\u09b0\u09be\u0982 AB \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3,\u00a0<\/span><\/p>\n\\frac{x}{p \\sec \\alpha}+\\frac{y}{p \\operatorname{cosec} \\alpha}=1<\/span>\n

\u09ac\u09be, \\frac{x \\cos \\alpha}{p}+\\frac{y \\sin \\alpha}{p}=1<\/span><\/span><\/p>\n

\u09ac\u09be, <\/span>x \\cos \\alpha+y \\sin \\alpha=p<\/span><\/span> \u0987\u09b9\u09be\u0987 AB \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u0964\u00a0\u00a0<\/span><\/p>\n

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\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