{"id":3645,"date":"2024-01-30T12:30:06","date_gmt":"2024-01-30T06:30:06","guid":{"rendered":"https:\/\/stage-wp.10minuteschool.com\/?p=3645"},"modified":"2024-10-29T15:07:31","modified_gmt":"2024-10-29T09:07:31","slug":"straight-line","status":"publish","type":"post","link":"https:\/\/10minuteschool.com\/content\/straight-line\/","title":{"rendered":"\u09b8\u09b0\u09b2 \u09b0\u09c7\u0996\u09be (Straight Line)"},"content":{"rendered":"<h2><span style=\"color: #339966;\"><b>\u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995 \u0995\u09be\u0995\u09c7 \u09ac\u09b2\u09c7? \u098f\u09ac\u0982 \u098f\u09b0 \u09ac\u09bf\u09b8\u09cd\u09a4\u09be\u09b0\u09bf\u09a4 \u09aa\u09cd\u09b0\u0995\u09be\u09b0\u09ad\u09c7\u09a6<\/b><\/span><\/h2>\n<h3><span style=\"color: #800080;\"><b>\u09b8\u09ae\u09a4\u09b2\u09c7 \u09aa\u09cb\u09b2\u09be\u09b0 \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995 \u0993 \u0995\u09be\u09b0\u09cd\u09a4\u09c7\u09b8\u09c0\u09af\u09bc \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995 <\/b><b>(Cartesian and Polar co-ordinates on plane):<\/b><\/span><\/h3>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><b><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/academic\/10\/?group=science\">\u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995<\/a><\/span> \u0995\u09be\u0995\u09c7 \u09ac\u09b2\u09c7? <\/b><b>(<\/b><b>Co-ordinates):<\/b> <span style=\"font-weight: 400;\">\u0995\u09cb\u09a8 \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09ac\u09be \u09ac\u09b8\u09cd\u09a4\u09c1\u09b0 \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09a8 \u09b8\u09c1\u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f\u09ad\u09be\u09ac\u09c7 \u09a8\u09bf\u09b0\u09cd\u09a3\u09af\u09bc\u09c7\u09b0 \u099c\u09a8\u09cd\u09af \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995 \u09ac\u09cd\u09af\u09ac\u09b9\u09be\u09b0 \u0995\u09b0\u09be \u09b9\u09af\u09bc\u0964 \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995 \u09ac\u09cd\u09af\u09ac\u09b8\u09cd\u09a5\u09be \u09a6\u09c1&#8217;\u09a7\u09b0\u09a8\u09c7\u09b0\u0964 \u09af\u09a5\u09be- <\/span><\/li>\n<\/ul>\n<p><b>i) \u0986\u09af\u09bc\u09a4 \u09ac\u09be \u0995\u09be\u09b0\u09cd\u09a4\u09c7\u09b8\u09c0\u09af\u09bc \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995 (<\/b><b>Rectangular\/Cartesian Co-ordinate)<\/b><b> \u09ac\u09cd\u09af\u09ac\u09b8\u09cd\u09a5\u09be: <\/b><span style=\"font-weight: 400;\">\u09aa\u09b0\u09b8\u09cd\u09aa\u09b0\u099a\u09cd\u099b\u09c7\u09a6\u09c0 \u09b2\u09ae\u09cd\u09ac\u09ad\u09be\u09ac\u09c7 \u0985\u09ac\u09b8\u09cd\u09a5\u09bf\u09a4 \u09a6\u09c1\u099f\u09bf \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u0995\u09c7 \u09a8\u09bf\u09af\u09bc\u09c7 \u09af\u09c7 \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995 \u09ac\u09cd\u09af\u09ac\u09b8\u09cd\u09a5\u09be \u09a4\u09be\u0995\u09c7 \u0995\u09be\u09b0\u09cd\u09a4\u09c7\u09b8\u09c0\u09af\u09bc \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995 \u09ac\u09cd\u09af\u09ac\u09b8\u09cd\u09a5\u09be \u09ac\u09b2\u09c7\u0964 \u098f\u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7 \u0985\u09a8\u09c1\u09ad\u09c2\u09ae\u09bf\u0995 \u09b0\u09c7\u0996\u09be\u0995\u09c7 x- \u0985\u0995\u09cd\u09b7 \u098f\u09ac\u0982 \u0989\u09b2\u09ae\u09cd\u09ac\u09b0\u09c7\u0996\u09be\u0995\u09c7 y-\u0985\u0995\u09cd\u09b7 \u09a7\u09b0\u09be \u09b9\u09af\u09bc \u098f\u09ac\u0982 \u0985\u0995\u09cd\u09b7 \u09a6\u09c1\u099f\u09bf\u09b0 \u099b\u09c7\u09a6\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u0995\u09c7 \u09ae\u09c2\u09b2\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09ac\u09b2\u09c7\u0964 \u098f\u0995\u099f\u09bf \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09b0 \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995 P(x, y) \u09ac\u09b2\u09a4\u09c7 \u09ac\u09c1\u099d\u09be\u09af\u09bc, \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u099f\u09bf x-\u0985\u0995\u09cd\u09b7 \u09a5\u09c7\u0995\u09c7 y- \u09b2\u09ae\u09cd\u09ac \u09a6\u09c2\u09b0\u09a4\u09cd\u09ac\u09c7 \u098f\u09ac\u0982 y-\u0985\u0995\u09cd\u09b7 \u09a5\u09c7\u0995\u09c7 x-\u09b2\u09ae\u09cd\u09ac \u09a6\u09c2\u09b0\u09a4\u09cd\u09ac\u09c7 \u0985\u09ac\u09b8\u09cd\u09a5\u09bf\u09a4\u0964<\/span><\/p>\n<p><b>ii) \u09aa\u09cb\u09b2\u09be\u09b0 \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995 (<\/b><b>Polar Co-ordinate)<\/b><b> \u09ac\u09cd\u09af\u09ac\u09b8\u09cd\u09a5\u09be: <\/b><span style=\"font-weight: 400;\">\u098f\u0995\u099f\u09bf \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u098f\u09ac\u0982 \u0990 \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u0997\u09be\u09ae\u09c0 \u098f\u0995\u099f\u09bf \u09b0\u09c7\u0996\u09be\u0995\u09c7 \u09a8\u09bf\u09af\u09bc\u09c7 \u09af\u09c7 \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995 \u09ac\u09cd\u09af\u09ac\u09b8\u09cd\u09a5\u09be \u09a4\u09be\u0995\u09c7 \u09aa\u09cb\u09b2\u09be\u09b0 \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995 \u09ac\u09b2\u09c7\u0964 \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u099f\u09bf\u0995\u09c7 \u09aa\u09cb\u09b2 \u09ac\u09be \u09ae\u09c2\u09b2\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u098f\u09ac\u0982 \u09b0\u09c7\u0996\u09be\u099f\u09bf\u0995\u09c7 \u0986\u09a6\u09bf\u09b0\u09c7\u0996\u09be (Initial line) \u09ac\u09be \u09aa\u09cb\u09b2\u09be\u09b0 \u0985\u0995\u09cd\u09b7 \u09ac\u09b2\u09c7\u0964<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><b>\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09b0 \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09a8 \u09a8\u09bf\u09b0\u09cd\u09a6\u09c7\u09b6: <\/b><span style=\"font-weight: 400;\">\u0995\u09be\u09b0\u09cd\u09a4\u09c7\u09b8\u09c0\u09af\u09bc \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995 \u09ac\u09cd\u09af\u09ac\u09b8\u09cd\u09a5\u09be\u09af\u09bc \u0995\u09cb\u09a8 \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09b0 \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09a8\u0995\u09c7 (x, y) \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09aa\u09cd\u09b0\u0995\u09be\u09b6 \u0995\u09b0\u09be \u09b9\u09af\u09bc\u0964 x \u0995\u09c7 \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u099f\u09bf\u09b0 \u09ad\u09c1\u099c (abscissa) \u098f\u09ac\u0982 y \u0995\u09c7 \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u099f\u09bf\u09b0 \u0995\u09cb\u099f\u09bf (ordinate) \u09ac\u09b2\u09c7\u0964 \u09aa\u09cb\u09b2\u09be\u09b0 \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995 \u09ac\u09cd\u09af\u09ac\u09b8\u09cd\u09a5\u09be\u09af\u09bc \u0995\u09cb\u09a8\u09cb \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09b0 \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09a8\u0995\u09c7 (r, \u03b8 <\/span><span style=\"font-weight: 400;\">) \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09aa\u09cd\u09b0\u0995\u09be\u09b6 \u0995\u09b0\u09be \u09b9\u09af\u09bc\u0964 r \u0995\u09c7 \u09ac\u09cd\u09af\u09be\u09b8\u09be\u09b0\u09cd\u09a7 \u09ad\u09c7\u0995\u09cd\u099f\u09b0 (Radius Vector) \u098f\u09ac\u0982 \u03b8<\/span><span style=\"font-weight: 400;\"> \u0995\u09c7 \u09ad\u09c7\u0995\u09cd\u099f\u09cb\u09b0\u09bf\u09af\u09bc\u09be\u09b2 \u0995\u09cb\u09a3 \u09ac\u09b2\u09c7\u0964<\/span><\/p>\n<ul>\n<li><b>\u09aa\u09cb\u09b2\u09be\u09b0 \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995 \u0993 \u0995\u09be\u09b0\u09cd\u09a4\u09c7\u09b8\u09c0\u09af\u09bc \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995\u09c7\u09b0 \u09ae\u09a7\u09cd\u09af\u09c7 \u09b8\u09ae\u09cd\u09aa\u09b0\u09cd\u0995 <\/b><b>(Relation between Polar and Cartesian co ordinates) <\/b><b>:\u00a0<\/b><\/li>\n<\/ul>\n<p><span style=\"font-weight: 400;\">\u099a\u09bf\u09a4\u09cd\u09b0\u09c7 P \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09b0 \u0995\u09be\u09b0\u09cd\u09a4\u09c7\u09b8\u09c0\u09af\u09bc \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995 (x, y) \u098f\u09ac\u0982 \u09aa\u09cb\u09b2\u09be\u09b0 \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995 (r, \u03b8<\/span><span style=\"font-weight: 400;\">) \u09b9\u09b2\u09c7 \u099a\u09bf\u09a4\u09cd\u09b0\u09be\u09a8\u09c1\u09b8\u09be\u09b0\u09c7 \u0986\u09ae\u09b0\u09be \u09aa\u09be\u0987, x = r cos\u03b8<\/span><span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">\u098f\u09ac\u0982 y = r sin\u03b8<\/span><\/p>\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\therefore r=\\sqrt{\\chi_{2}+y_{2}} <\/span>\u00a0 <\/span><span style=\"font-weight: 400;\">\u098f\u09ac\u0982<\/span><span style=\"font-weight: 400;\"> <span class=\"katex-eq\" data-katex-display=\"false\"> \\tan \\theta = \\frac{y}{x} <\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">P(x,y) \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u099f\u09bf\u09b0 \u099a\u09be\u09b0\u099f\u09bf \u099a\u09a4\u09c1\u09b0\u09cd\u09ad\u09be\u0997\u09c7 \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09a8\u09c7\u09b0 \u0989\u09aa\u09b0 \u09ad\u09bf\u09a4\u09cd\u09a4\u09bf \u0995\u09b0\u09c7 \u0986\u09b0\u09cd\u0997\u09c1\u09ae\u09c7\u09a8\u09cd\u099f (\u03b8<\/span><span style=\"font-weight: 400;\">) \u09a8\u09bf\u09b0\u09cd\u09a3\u09af\u09bc\u09c7\u09b0 \u09b8\u09be\u09a7\u09be\u09b0\u09a3 \u099a\u09be\u09b0\u099f\u09bf \u09a8\u09bf\u09af\u09bc\u09ae:<\/span><\/p>\n<ol>\n<li><span style=\"font-weight: 400;\"> P(x,y) \u098f\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7; <span class=\"katex-eq\" data-katex-display=\"false\"> \\theta = \\tan^{-1} \\theta = \\frac{y}{x} <\/span><\/span><\/li>\n<li><span style=\"font-weight: 400;\"> P(-x,y) \u098f\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7; <span class=\"katex-eq\" data-katex-display=\"false\"> \\theta = \\pi - \\tan^{-1} \\theta = \\frac{y}{x} <\/span><\/span><\/li>\n<li><span style=\"font-weight: 400;\"> P(-x,-y) \u098f\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7; <span class=\"katex-eq\" data-katex-display=\"false\"> \\theta = \\pi + \\tan^{-1} \\theta = \\frac{y}{x} <\/span><\/span><b>\u00a0<\/b><span style=\"font-weight: 400;\">;\u09af\u0996\u09a8, 0 <\/span><span style=\"font-weight: 400;\">\u2264\u00a0\u03b8\u00a0&lt;2\u03c0<\/span><\/li>\n<\/ol>\n<p><span style=\"font-weight: 400;\">Or, <\/span><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\theta = \\pi + \\tan^{-1} \\theta = \\frac{y}{x} <\/span><\/span><b> \u00a0<\/b><span style=\"font-weight: 400;\">;\u09af\u0996\u09a8, -\u03c0<\/span> <span style=\"font-weight: 400;\">&lt;\u00a0\u03b8\u00a0\u2264\u03c0<\/span><\/p>\n<ol start=\"4\">\n<li><span style=\"font-weight: 400;\"> P(x,-y) \u098f\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7; <\/span><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\theta = 2 \\pi + \\tan^{-1} \\theta = \\frac{y}{x} <\/span><\/span><b> <\/b><span style=\"font-weight: 400;\">;\u09af\u0996\u09a8, 0 <\/span><span style=\"font-weight: 400;\">\u2264\u00a0\u03b8\u00a0&lt;2\u03c0<\/span><\/li>\n<\/ol>\n<p><span style=\"font-weight: 400;\">Or, <span class=\"katex-eq\" data-katex-display=\"false\">\\theta = - \\tan^{-1} \\theta = \\frac{x}{y} <\/span><\/span><b>\u00a0<\/b><span style=\"font-weight: 400;\">; \u09af\u0996\u09a8, -\u03c0<\/span> <span style=\"font-weight: 400;\">&lt;\u00a0\u03b8\u00a0\u2264\u03c0<\/span><\/p>\n<h3><span style=\"color: #800080;\"><b>\u09a6\u09c1\u0987\u099f\u09bf \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09b0 \u09ae\u09a7\u09cd\u09af\u0995\u09be\u09b0 \u09a6\u09c2\u09b0\u09a4\u09cd\u09ac <\/b><b>(Distance between two points)<\/b><b>:<\/b><\/span><\/h3>\n<p><b>i) \u0995\u09be\u09b0\u09cd\u09a4\u09c7\u09b8\u09c0\u09df \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0982\u0995<\/b><\/p>\n<p><span style=\"font-weight: 400;\">P (x<\/span><span style=\"font-weight: 400;\">1<\/span><span style=\"font-weight: 400;\">, y<\/span><span style=\"font-weight: 400;\">1<\/span><span style=\"font-weight: 400;\">) \u0993 Q (x<\/span><span style=\"font-weight: 400;\">2<\/span><span style=\"font-weight: 400;\">, y<\/span><span style=\"font-weight: 400;\">2<\/span><span style=\"font-weight: 400;\">) \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09a6\u09c1\u0987\u099f\u09bf\u09b0 \u09ae\u09a7\u09cd\u09af\u0995\u09be\u09b0 \u09a6\u09c2\u09b0\u09a4\u09cd\u09ac \u09a8\u09bf\u09b0\u09cd\u09a3\u09af\u09bc \u0995\u09b0\u09a4\u09c7 \u09b9\u09ac\u09c7\u0964 x \u0985\u0995\u09cd\u09b7\u09c7\u09b0 \u0989\u09aa\u09b0 PL, OM \u09b2\u09ae\u09cd\u09ac \u099f\u09be\u09a8\u09bf \u098f\u09ac\u0982 OM \u098f\u09b0 \u0989\u09aa\u09b0 PV \u09b2\u09ae\u09cd\u09ac \u099f\u09be\u09a8\u09bf\u0964<\/span><\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\"> \\therefore \\mathrm{OL}=\\mathrm{x}_{1}, \\mathrm{OM}=\\mathrm{x}_{2}, \\mathrm{NM}=\\mathrm{PL}=\\mathrm{y}_{1}, \\mathrm{QM}=\\mathrm{y}_{2} <\/span>\n<p><span style=\"font-weight: 400;\">\u2234<\/span><span style=\"font-weight: 400;\"> P,Q \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a6\u09cd\u09ac\u09df\u09c7\u09b0 \u09a6\u09c2\u09b0\u09a4\u09cd\u09ac = PQ<\/span><\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\"> = \\sqrt{P N^{2}+Q N^{2}} <\/span>\n<p>&nbsp;<\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\"> = \\sqrt{(LM)^{2} + (QM - NM)^{2}} <\/span>\n<p>&nbsp;<\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\"> = \\sqrt{(OM - OL)^{2} + (QM - PL)^{2}} <\/span>\n<p>&nbsp;<\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\"> = \\sqrt{(x_2 - x_1)^{2} + (y_2 - y_1)^{2}} <\/span>\n<p>&nbsp;<\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\"> =\\sqrt{(\\text { \u09a4\u09c1\u099c\u09a6\u09cd\u09ac\u09af\u09bc\u09c7\u09b0 \u09aa\u09be\u09b0\u09cd\u09a5\u0995\u09cd\u09af) })^{2}+(\\text { \u0995\u09cb\u099f\u09bf\u09a6\u09cd\u09ac\u09af\u09bc\u09c7\u09b0 \u09aa\u09be\u09b0\u09cd\u09a5\u0995\u09cd\u09af })^{2}} <\/span>\n<p>&nbsp;<\/p>\n<p><b>(ii) \u09aa\u09cb\u09b2\u09be\u09b0 \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995: <\/b><span style=\"font-weight: 400;\">\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, O \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09ae\u09c2\u09b2\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u0993 OX \u0986\u09a6\u09bf \u09b0\u09c7\u0996\u09be \u09b8\u09be\u09aa\u09c7\u0995\u09cd\u09b7\u09c7 \u09a6\u09c1\u0987\u099f\u09bf \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 <span class=\"katex-eq\" data-katex-display=\"false\"> P_1 (r_1, \\theta_1) <\/span> <\/span><span style=\"font-weight: 400;\">\u098f\u09ac\u0982 <span class=\"katex-eq\" data-katex-display=\"false\"> P_2 (r_2, \\theta_2) <\/span><\/span><span style=\"font-weight: 400;\">\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09b8\u09c1\u09a4\u09b0\u09be\u0982, <span class=\"katex-eq\" data-katex-display=\"false\"> P_1 (r_1, \\theta_1) <\/span><\/span><span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09b0 \u0995\u09be\u09b0\u09cd\u09a4\u09c7\u09b8\u09c0\u09af\u09bc \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995 <span class=\"katex-eq\" data-katex-display=\"false\">(r_1 \\cos \\theta_1 , r_1 \\sin \\theta_1)\u00a0 <\/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\"> P_2 (r_2, \\theta_2) <\/span>\u00a0<\/span><span style=\"font-weight: 400;\">\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09b0 \u0995\u09be\u09b0\u09cd\u09a4\u09c7\u09b8\u09c0\u09af\u09bc \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995 <span class=\"katex-eq\" data-katex-display=\"false\">(r_2 \\cos \\theta_2 , r_2 \\sin \\theta_2)\u00a0 <\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\"> P_1 <\/span> \u098f\u09ac\u0982 <span class=\"katex-eq\" data-katex-display=\"false\"> P_2 <\/span><\/span><span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">\u00a0\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a6\u09cd\u09ac\u09af\u09bc\u09c7\u09b0 \u09ae\u09a7\u09cd\u09af\u09ac\u09b0\u09cd\u09a4\u09c0 \u09a6\u09c2\u09b0\u09a4\u09cd\u09ac,<\/span><\/p>\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\"> P_1 P_2 = \\sqrt{(\u09ad\u09c1\u099c\u09a6\u09cd\u09ac\u09df\u09c7\u09b0\u00a0\u09aa\u09be\u09b0\u09cd\u09a5\u0995\u09cd\u09af)^{2} + (\u0995\u09cb\u099f\u09bf\u09a6\u09cd\u09ac\u09df\u09c7\u09b0\u00a0\u09aa\u09be\u09b0\u09cd\u09a5\u0995\u09cd\u09af)^{2}}<\/span><\/span><\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\"> = \\sqrt{\\left(r_{2} \\cos \\theta_{2}-r_{1} \\cos \\theta_{1}\\right)^{2}+\\left(r_{2} \\sin \\theta_{2}-r_{1} \\sin \\theta_{1}\\right)^{2}} <\/span>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\sqrt{r_{2}^{2} \\cos ^{2} \\theta_{2}-2 r_{1} r_{2} \\cos \\theta_{2} \\cos \\theta_{1}+r_{1}^{2} \\cos ^{2} \\theta_{1}+r_{2}^{2} \\sin ^{2} \\theta_{2}-2 r_{1} r_{2} \\sin \\theta_{2} \\sin \\theta_{1}+r_{1}^{2} \\sin ^{2} \\theta_{1}}\u00a0 <\/span>\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\"> =\\sqrt{r_{2}^{2}\\left(\\cos ^{2} \\theta_{2}+\\sin ^{2} \\theta_{2}\\right)+r_{1}^{2}\\left(\\cos ^{2} \\theta_{1}+\\sin ^{2} \\theta_{1}\\right)-2 r_{1} r_{2}\\left(\\cos \\theta_{2} \\cos \\theta_{1}+\\sin \\theta_{2} \\sin \\theta_{1}\\right)} <\/span>)<\/span><\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">=\\sqrt{r_{2}^{2}+r_{1}^{2}-2 r_{1} r_{2} \\cos \\left(\\theta_{2}-\\theta_{1}\\right)} \u00a0 <\/span>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\mathrm{P}_{1} \\mathrm{P}_{2}=\\sqrt{r_{2}^{2}+r_{1}^{2}-2 r_{1} r_{2} \\cos \\left(\\theta_{2}-\\theta_{1}\\right)}\u00a0 <\/span>\n<p><b>Note: <\/b><span class=\"katex-eq\" data-katex-display=\"false\">(r_1, \\theta_1)\u00a0 <\/span><span style=\"font-weight: 400;\"> \u098f\u09ac\u0982 <span class=\"katex-eq\" data-katex-display=\"false\">(r_2, \\theta_2)\u00a0 <\/span><\/span><span style=\"font-weight: 400;\">\u00a0\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a6\u09cd\u09ac\u09af\u09bc\u09c7\u09b0 \u09ae\u09a7\u09cd\u09af\u09ac\u09b0\u09cd\u09a4\u09c0, \u09a6\u09c2\u09b0\u09a4\u09cd\u09ac \u09a8\u09bf\u09b0\u09cd\u09a3\u09af\u09bc\u09c7\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7<\/span><\/p>\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\sqrt{r_{2}^{2}+r_{1}^{2}-2 r_{1} r_{2} \\cos \\left(\\theta_{2}-\\theta_{1}\\right)} \\text { \u098f\u09ac\u0982 } \\sqrt{r_{2}^{2}+r_{1}^{2}-2 r_{1} r_{2} \\cos \\left(\\theta_{1}-\\theta_{2}\\right)} <\/span> \u098f\u0995\u0987 \u09ae\u09be\u09a8 \u09a8\u09bf\u09b0\u09cd\u09a6\u09c7\u09b6 \u0995\u09b0\u09c7\u0964<\/span><\/p>\n<p>&nbsp;<\/p>\n<h3><span style=\"color: #800080;\"><b>\u09e9.\u09e7\u09ed \u098f\u0995\u099f\u09bf \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09b9\u09a4\u09c7 \u098f\u0995\u099f\u09bf \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09b0\u09c7\u0996\u09be\u09b0 \u09b2\u09ae\u09cd\u09ac \u09a6\u09c2\u09b0\u09a4\u09cd\u09ac <\/b><b>(Perpendicular distance from a fixed point to a fixed line)<\/b><\/span><\/h3>\n<p><span style=\"font-weight: 400;\">\u09a7\u09b0\u09bf, \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09b0\u09c7\u0996\u09be\u099f\u09bf\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3, <\/span><span style=\"font-weight: 400;\">ax\u00a0+\u00a0by\u00a0+\u00a0c\u00a0=\u00a00<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u098f\u09ac\u0982 <span class=\"katex-eq\" data-katex-display=\"false\"> (x_1, y_1) <\/span><\/span><span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09a5\u09c7\u0995\u09c7 \u098f \u09b0\u09c7\u0996\u09be\u09b0 \u0989\u09aa\u09b0 \u09b2\u09ae\u09cd\u09ac \u09a6\u09c2\u09b0\u09a4\u09cd\u09ac = p<\/span><\/p>\n<p><span style=\"font-weight: 400;\">a<\/span><span style=\"font-weight: 400;\">x<\/span><span style=\"font-weight: 400;\">+by+c=0<\/span><span style=\"font-weight: 400;\"> \u09b0\u09c7\u0996\u09be\u09b0 \u0989\u09aa\u09b0 \u09a6\u09c1\u0987\u099f\u09bf \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 <span class=\"katex-eq\" data-katex-display=\"false\"> \\mathrm{A}\\left(-\\frac{c}{a}, 0\\right) \\text { \u0e07 } \\mathrm{B}\\left(0, \\frac{-c}{b}\\right) <\/span><\/span><span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\"> \u09a8\u09bf\u0987\u0964\u00a0<\/span><\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\"> \\Delta \\mathrm{A} B P=\\frac{1}{2}\\left[\\begin{array}{ccc}\n\n\\frac{-c}{a} &amp; 0 &amp; 1 \\\\\n\n0 &amp; \\frac{-c}{b} &amp; 1 \\\\\n\nx_{1} &amp; y_{1} &amp; 1\n\n\\end{array}\\right] <\/span>\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\"> =\\frac{1}{2}\\left\\{\\frac{c}{a}\\left(y_{1}+\\frac{c}{b}\\right)+x_{1}\\left(0+\\frac{c}{b}\\right)\\right\\} <\/span><\/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\"> =\\frac{1}{2}\\left(\\frac{c y_{1}}{a}+\\frac{c^{2}}{a b}+\\frac{c x_{1}}{b}\\right) <\/span><\/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\"> =\\frac{c}{2}\\left(\\frac{b y_{1}+a x_{1}+c}{a b}\\right) <\/span><\/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\"> =\\frac{c}{2 a b}\\left(b y_{1}+a x_{1}+c\\right) <\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u2234<\/span><span style=\"font-weight: 400;\"> \u09b2\u09ae\u09cd\u09ac\u09c7\u09b0 \u09a6\u09c8\u09b0\u09cd\u0998\u09cd\u09af, <span class=\"katex-eq\" data-katex-display=\"false\"> \\mathrm{p}=\\frac{\\left|a x_{1}+b y_{1}+c\\right|}{\\sqrt{a^{2}+b^{2}}} <\/span> <\/span><span style=\"font-weight: 400;\">[\u09ac\u09bf\u09b6\u09c7\u09b7 \u0995\u09cb\u09a8 \u09af\u09c1\u0995\u09cd\u09a4\u09bf \u0989\u09b2\u09cd\u09b2\u09c7\u0996 \u09a8\u09be \u09a5\u09be\u0995\u09b2\u09c7 \u09b2\u09ae\u09cd\u09ac\u09c7\u09b0 \u09a6\u09c8\u09b0\u09cd\u0998\u09cd\u09af \u09a7\u09a8\u09be\u09a4\u09cd\u09ae\u0995 \u09a7\u09b0\u09be \u09b9\u09df\u0964]<\/span><\/p>\n<p><b>\u09b2\u09ae\u09cd\u09ac\u09c7\u09b0 \u099a\u09bf\u09b9\u09cd\u09a8\u0983 <\/b><span style=\"font-weight: 400;\">\u09af\u09a6\u09bf \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u099f\u09bf \u098f\u09ae\u09a8\u09ad\u09be\u09ac\u09c7 \u09b2\u09c7\u0996\u09be \u09af\u09be\u09df \u09af\u09c7, \u0985\u09a8\u09aa\u09c7\u0995\u09cd\u09b7 \u09aa\u09a6 (absolute term) \u09af\u09cb\u0997\u09ac\u09cb\u09a7\u0995 \u09b9\u09df, \u09a4\u09be\u09b9\u09b2\u09c7 p \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u098f\u09ac\u0982 \u09ae\u09c2\u09b2\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09b0\u09c7\u0996\u09be\u099f\u09bf\u09b0 \u098f\u0995\u0987 \u09aa\u09be\u09b0\u09cd\u09b6\u09cd\u09ac\u09c7 \u09aa\u09dc\u09b2\u09c7 \u09b2\u09ae\u09cd\u09ac \u09a7\u09a8\u09be\u09a4\u09cd\u09ae\u0995 \u09b9\u09ac\u09c7 \u098f\u09ac\u0982 \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u09aa\u09be\u09b0\u09cd\u09b6\u09cd\u09ac\u09c7 \u09aa\u09dc\u09b2\u09c7 \u09b2\u09ae\u09cd\u09ac \u098b\u09a3\u09be\u09a4\u09cd\u09ae\u0995 \u09b9\u09ac\u09c7\u0964\u00a0<\/span><\/p>\n<p>&nbsp;<\/p>\n<h3><span style=\"color: #800080;\"><b>\u09e9.\u09e7\u09ed.\u09e7 \u0995\u09cb\u09a8 \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09b9\u09a4\u09c7 \u0995\u09cb\u09a8\u09cb \u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09ae\u09be\u09a8\u09cd\u09a4\u09b0\u09be\u09b2 \u09a6\u09bf\u0995\u09c7 \u0985\u09aa\u09b0 \u0995\u09cb\u09a8\u09cb \u09b0\u09c7\u0996\u09be\u09b0 \u09a6\u09c2\u09b0\u09a4\u09cd\u09ac\u00a0<\/b><\/span><\/h3>\n<p><span style=\"font-weight: 400;\">(<\/span><b>Perpendicular distance from a fixed point to a fixed line)<\/b><\/p>\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\"> A(x_1, y_1) <\/span> <\/span><span style=\"font-weight: 400;\">\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09b9\u09a4\u09c7 <span class=\"katex-eq\" data-katex-display=\"false\"> a_1 x+b_1 y+c_1 = 0 <\/span> <\/span><span style=\"font-weight: 400;\">\u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09ae\u09be\u09a8\u09cd\u09a4\u09b0\u09be\u09b2 \u09a6\u09bf\u0995\u09c7\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\"> a_2 x+b_2 y+c_2 = 0 <\/span> \u09b0\u09c7\u0996\u09be\u09b0 \u09a6\u09c2\u09b0\u09a4\u09cd\u09ac \u09a8\u09bf\u09b0\u09cd\u09a3\u09df\u09c7\u09b0 \u099c\u09a8\u09cd\u09af, \u09aa\u09cd\u09b0\u09a5\u09ae\u09c7\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\"> a_1 x+b_1 y+c_1 = 0 <\/span> <\/span><span style=\"font-weight: 400;\">\u098f\u09b0 \u09b8\u09ae\u09be\u09a8\u09cd\u09a4\u09b0\u09be\u09b2 \u09af\u09c7 \u0995\u09cb\u09a8\u09cb \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\"> a_1 x+b_1 y+c_1 = 0 <\/span> <\/span><span style=\"font-weight: 400;\">\u09a8\u09bf\u0987\u0964 \u098f\u0987 \u09b0\u09c7\u0996\u09be\u09b0 \u0989\u09aa\u09b0 <span class=\"katex-eq\" data-katex-display=\"false\"> A(x_1, y_1) <\/span> <\/span><span style=\"font-weight: 400;\">\u0985\u09ac\u09b8\u09cd\u09a5\u09bf\u09a4\u0964\u00a0\u00a0\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\therefore a_1 x+b_1 y+c_1 = 0 <\/span> <\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09ac\u09be, <span class=\"katex-eq\" data-katex-display=\"false\"> k = -a_1 x - b_1 y <\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u098f\u0996\u09a8 <span class=\"katex-eq\" data-katex-display=\"false\"> a_2 x+b_2 y+c_2 = 0 <\/span><\/span><span style=\"font-weight: 400;\"> \u098f\u09ac\u0982 <span class=\"katex-eq\" data-katex-display=\"false\"> a_1 x+b_1 y+ k\u00a0 = 0 <\/span> <\/span><span style=\"font-weight: 400;\">\u00a0\u09b8\u09ae\u09be\u09a7\u09be\u09a8 \u0995\u09b0\u09c7 \u099b\u09c7\u09a6\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 <span class=\"katex-eq\" data-katex-display=\"false\"> B(x_2, y_2) <\/span><\/span><span style=\"font-weight: 400;\">\u00a0\u09aa\u09be\u0993\u09df\u09be \u09af\u09be\u09ac\u09c7\u0964\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09a4\u09be\u09b9\u09b2\u09c7, \u09a8\u09bf\u09b0\u09cd\u09a3\u09c7\u09df \u09a6\u09c2\u09b0\u09a4\u09cd\u09ac <span class=\"katex-eq\" data-katex-display=\"false\"> \\mathrm{AB}=\\sqrt{\\left(x_{1}-x_{2}\\right)^{2}+\\left(y_{1}-y_{2}\\right)^{2}} <\/span><\/span><\/p>\n<p>&nbsp;<\/p>\n<h3><span style=\"color: #800080;\"><b>\u09e9.\u09e7\u09ed.\u09e8 \u00a0 \u09a6\u09c1\u0987\u099f\u09bf \u09b8\u09ae\u09be\u09a8\u09cd\u09a4\u09b0\u09be\u09b2 \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u09ae\u09a7\u09cd\u09af\u09ac\u09b0\u09cd\u09a4\u09c0 \u09b2\u09ae\u09cd\u09ac \u09a6\u09c2\u09b0\u09a4\u09cd\u09ac <\/b><b>(Perpendicular distance of two parallel <span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/www.youtube.com\/watch?v=cB0DqOSL26M&amp;list=PL1pf33qWCkmidHi8s0NpGydwZAj3fGIhQ\" target=\"_blank\" rel=\"noopener\">straight lines<\/a><\/span>)<\/b><\/span><\/h3>\n<p><span style=\"font-weight: 400;\">\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf<\/span><span style=\"font-weight: 400;\">, <\/span><span style=\"font-weight: 400;\">\u09b8\u09ae\u09be\u09a8\u09cd\u09a4\u09b0\u09be\u09b2 \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be \u09a6\u09c1\u0987\u099f\u09bf <\/span><span style=\"font-weight: 400;\">AB\u00a0<\/span><span style=\"font-weight: 400;\">\u0993<\/span><span style=\"font-weight: 400;\">\u00a0CD\u00a0<\/span><span style=\"font-weight: 400;\">\u098f\u09b0<\/span><span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">\u09b8\u09ae\u09c0\u0995\u09b0\u09a3<\/span><span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">\u09af\u09a5\u09be\u0995\u09cd\u09b0\u09ae\u09c7\u00a0<\/span><span style=\"font-weight: 400;\"> <span class=\"katex-eq\" data-katex-display=\"false\"> a_1 x+b_1 y+c_1 = 0 <\/span> <\/span>\u00a0<span style=\"font-weight: 400;\">\u098f\u09ac\u0982<\/span> <span class=\"katex-eq\" data-katex-display=\"false\"> ax+by+c_1 = 0 <\/span> <span style=\"font-weight: 400;\">.<\/span><\/p>\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\"> E(x_1, y_1) <\/span> <\/span><span style=\"font-weight: 400;\">\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09b9\u09a4\u09c7<\/span><span style=\"font-weight: 400;\"> AB <\/span><span style=\"font-weight: 400;\">\u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u0989\u09aa\u09b0 \u0985\u09ac\u09b8\u09cd\u09a5\u09bf\u09a4\u0964<\/span><span style=\"font-weight: 400;\">\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\therefore a \\times_{1}+b y_{1}+c_{1}=0 \\ldots \\quad \\ldots \\quad \\ldots \\text { (i) } <\/span>\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">E\u00a0<\/span><span style=\"font-weight: 400;\">\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09b9\u09a4\u09c7<\/span><span style=\"font-weight: 400;\"> CD <\/span><span style=\"font-weight: 400;\">\u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u0989\u09aa\u09b0<\/span><span style=\"font-weight: 400;\"> EF <\/span><span style=\"font-weight: 400;\">\u09b2\u09ae\u09cd\u09ac \u0986\u0981\u0995\u09be \u09b9\u09b2\u0964<\/span><span style=\"font-weight: 400;\">\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">E(<\/span><span style=\"font-weight: 400;\">x<\/span><span style=\"font-weight: 400;\">1<\/span><span style=\"font-weight: 400;\">,<\/span><span style=\"font-weight: 400;\">y<\/span><span style=\"font-weight: 400;\">1)<span class=\"katex-eq\" data-katex-display=\"false\"> E(x_1, y_1) <\/span> <\/span><span style=\"font-weight: 400;\">\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1<\/span><span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">\u09b9\u09a4\u09c7<\/span><span style=\"font-weight: 400;\"> <span class=\"katex-eq\" data-katex-display=\"false\"> ax+by+c_2 = 0 <\/span> <\/span><span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">\u09b0\u09c7\u0996\u09be\u09b0<\/span><span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">\u0989\u09aa\u09b0<\/span><span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">\u09b2\u09ae\u09cd\u09ac<\/span><span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">\u09a6\u09c2\u09b0\u09a4\u09cd\u09ac<\/span><span style=\"font-weight: 400;\">, <span class=\"katex-eq\" data-katex-display=\"false\"> E F=\\frac{\\left|a x_{1}+b y_{1}+c_{2}\\right|}{\\sqrt{a^{2}+b^{2}}} <\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf<\/span><span style=\"font-weight: 400;\">, <\/span><span style=\"font-weight: 400;\">P\u00a0=\u00a0EF<\/span><\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\"> \\therefore p = \\frac{\\left|a x_{1}+b y_{1}+c_{2}\\right|}{\\sqrt{a^{2}+b^{2}}} <\/span>\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\">\u00a0 = \\frac{\\left| - c_1 + c_{2}\\right|}{\\sqrt{a^{2}+b^{2}}} <\/span> <\/span><span style=\"font-weight: 400;\">[<\/span><span style=\"font-weight: 400;\">\u09b8\u09ae\u09c0\u0995\u09b0\u09a3<\/span><span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">i<\/span><span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">\u09a8\u0982<\/span><span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">\u09a5\u09c7\u0995\u09c7<\/span><span style=\"font-weight: 400;\"> <span class=\"katex-eq\" data-katex-display=\"false\"> ax_1 + by_1 = -c_1 <\/span><\/span><span style=\"font-weight: 400;\">]<\/span><\/p>\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\">\u00a0 = \\frac{\\left| c_{2} - c_1 \\right|}{\\sqrt{a^{2}+b^{2}}} <\/span> <\/span><\/p>\n<p><span style=\"font-weight: 400;\">AB <\/span><span style=\"font-weight: 400;\">\u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3<\/span> <span class=\"katex-eq\" data-katex-display=\"false\"> ax+by+c_2 = 0 <\/span><span style=\"font-weight: 400;\">\u00a0\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u098f\u09ac\u0982<\/span><span style=\"font-weight: 400;\"> CD-<\/span><span style=\"font-weight: 400;\">\u098f\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3<\/span>\u00a0<span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\"> ax+by+c_1 = 0 <\/span>\u00a0<\/span><span style=\"font-weight: 400;\">\u09b9\u09b2\u09c7<\/span><span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">\u09b8\u09ae\u09be\u09a8\u09cd\u09a4\u09b0\u09be\u09b2<\/span><span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">\u09b0\u09c7\u0996\u09be<\/span><span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">\u09a6\u09c1\u0987\u099f\u09bf\u09b0<\/span><span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">\u09ae\u09a7\u09cd\u09af\u09ac\u09b0\u09cd\u09a4\u09c0<\/span><span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">\u09b2\u09ae\u09cd\u09ac<\/span><span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">\u09a6\u09c2\u09b0\u09a4\u09cd\u09ac<\/span><span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">\u09b9\u09ac\u09c7 <span class=\"katex-eq\" data-katex-display=\"false\">\u00a0 \\frac{\\left| c_{2} - c_1 \\right|}{\\sqrt{a^{2}+b^{2}}} <\/span>.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09b8\u09c1\u09a4\u09b0\u09be\u0982<\/span> <span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\"> ax+by+c_1 = 0 <\/span>\u00a0<\/span>\u00a0<span style=\"font-weight: 400;\">\u098f\u09ac\u0982<\/span> <span class=\"katex-eq\" data-katex-display=\"false\"> ax+by+c_2 = 0 <\/span><span style=\"font-weight: 400;\">\u00a0\u00a0<\/span>\u00a0<span style=\"font-weight: 400;\">\u09b8\u09ae\u09be\u09a8\u09cd\u09a4\u09b0\u09be\u09b2 \u09b0\u09c7\u0996\u09be\u09a6\u09cd\u09ac\u09df\u09c7\u09b0 \u09ae\u09a7\u09cd\u09af\u09ac\u09b0\u09cd\u09a4\u09c0<\/span> <span style=\"font-weight: 400;\">\u09a6\u09c2\u09b0\u09a4\u09cd\u09ac<\/span><span style=\"font-weight: 400;\">\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\">\u00a0 = \\frac{\\left| c_{2} - c_1 \\right|}{\\sqrt{a^{2}+b^{2}}} = = \\frac{\\left| c_1 - c_2\\right|}{\\sqrt{a^{2}+b^{2}}} <\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u2234<\/span> <span style=\"font-weight: 400;\">\u09a6\u09c1\u0987\u099f\u09bf<\/span> <span style=\"font-weight: 400;\">\u09b8\u09ae\u09be\u09a8\u09cd\u09a4\u09b0\u09be\u09b2<\/span> <span style=\"font-weight: 400;\">\u09b0\u09c7\u0996\u09be\u09a6\u09cd\u09ac\u09df\u09c7\u09b0<\/span> <span style=\"font-weight: 400;\">\u09ae\u09a7\u09cd\u09af\u09ac\u09b0\u09cd\u09a4\u09c0<\/span> <span style=\"font-weight: 400;\">\u09a6\u09c2\u09b0\u09a4\u09cd\u09ac<\/span> <span class=\"katex-eq\" data-katex-display=\"false\"> \\frac{| \\text{\u09a7\u09cd\u09b0\u09c1\u09ac\u0995\u09a6\u09cd\u09ac\u09df\u09c7\u09b0 \u0985\u09a8\u09cd\u09a4\u09b0} |}{\\sqrt{(x \\text{\u098f\u09b0 \u09b8\u09b9\u0997})^2 + (y \\text{\u098f\u09b0 \u09b8\u09b9\u0997})^2}} <\/span><\/p>\n<p>&nbsp;<\/p>\n<h3><span style=\"color: #800080;\"><b>\u09e9.\u09e7\u09ed.\u09e9 \u00a0 \u09a6\u09c1\u0987\u099f\u09bf \u0985\u09b8\u09ae\u09be\u09a8\u09cd\u09a4\u09b0\u09be\u09b2 \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u09ae\u09a7\u09cd\u09af\u09ac\u09b0\u09cd\u09a4\u09c0 \u0995\u09cb\u09a3\u09c7\u09b0 \u09b8\u09ae\u09a6\u09cd\u09ac\u09bf\u0996\u09a3\u09cd\u09a1\u0995\u09c7\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 <\/b><b>(Equation of bisectors of the angles between two straight lines)<\/b><\/span><\/h3>\n<p><span style=\"font-weight: 400;\">\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, LM \u098f\u09ac\u0982 RS \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09a6\u09cd\u09ac\u09df A \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 \u099b\u09c7\u09a6 \u0995\u09b0\u09c7 \u098f\u09ac\u0982 \u098f\u09a6\u09c7\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u09af\u09a5\u09be\u0995\u09cd\u09b0\u09ae\u09c7,\u00a0<\/span><\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\"> a x_{1}+b y_{1}+c_{1}=0 \\ldots \\quad \\ldots \\quad \\ldots \\text { (i) } <\/span>\n<p><span style=\"font-weight: 400;\">\u098f\u09ac\u0982 <span class=\"katex-eq\" data-katex-display=\"false\"> a x_{2}+b y_{2}+c_{2}=0 \\ldots \\quad \\ldots \\quad \\ldots \\text { (ii) } <\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09a7\u09b0\u09bf, AD \u098f\u09ac\u0982 AE \u09b0\u09c7\u0996\u09be\u09a6\u09cd\u09ac\u09af\u09bc LM \u098f\u09ac\u0982 RS \u09b0\u09c7\u0996\u09be\u09a6\u09cd\u09ac\u09af\u09bc\u09c7\u09b0 \u09ae\u09a7\u09cd\u09af\u09ac\u09b0\u09cd\u09a4\u09c0 \u0995\u09cb\u09a3\u09a6\u09cd\u09ac\u09af\u09bc\u09c7\u09b0 \u09b8\u09ae\u09a6\u09cd\u09ac\u09bf\u0996\u09a3\u09cd\u09a1\u0995\u09a6\u09cd\u09ac\u09af\u09bc\u0964 (i) \u0993 (ii) \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09a6\u09cd\u09ac\u09af\u09bc\u0995\u09c7 \u098f\u09ae\u09a8\u09ad\u09be\u09ac\u09c7 \u09b2\u09bf\u0996\u09be \u09b9\u09af\u09bc\u09c7\u099b\u09c7 \u09af\u09c7\u09a8 <span class=\"katex-eq\" data-katex-display=\"false\"> C_1 <\/span><\/span><span style=\"font-weight: 400;\"> \u098f\u09ac\u0982 <span class=\"katex-eq\" data-katex-display=\"false\"> C_2 <\/span><\/span><span style=\"font-weight: 400;\"> \u0989\u09ad\u09af\u09bc\u0987 \u09af\u09cb\u0997\u09ac\u09cb\u09a7\u0995\u0964 \u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, RAL \u0995\u09cb\u09a3\u09c7\u09b0 \u09b8\u09ae\u09a6\u09cd\u09ac\u09bf\u0996\u09a3\u09cd\u09a1\u0995 AD \u09b0\u09c7\u0996\u09be\u09b0 \u0989\u09aa\u09b0 <span class=\"katex-eq\" data-katex-display=\"false\"> P(x_1, y_2) <\/span><\/span><span style=\"font-weight: 400;\"> \u09af\u09c7 \u0995\u09cb\u09a8\u09cb \u098f\u0995\u099f\u09bf \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u0964 P \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09b9\u09a4\u09c7 LM \u098f\u09ac\u0982 RS \u09b0\u09c7\u0996\u09be\u09a6\u09cd\u09ac\u09af\u09bc\u09c7\u09b0 \u0989\u09aa\u09b0 \u09af\u09a5\u09be\u0995\u09cd\u09b0\u09ae\u09c7 <span class=\"katex-eq\" data-katex-display=\"false\"> PM_1 <\/span><\/span><span style=\"font-weight: 400;\"> \u098f\u09ac\u0982 <span class=\"katex-eq\" data-katex-display=\"false\"> PM_2 <\/span><\/span><span style=\"font-weight: 400;\"> \u09b2\u09ae\u09cd\u09ac \u099f\u09be\u09a8\u09bf\u0964 \u098f\u0996\u09a8 \u09af\u09c7\u09b9\u09c7\u09a4\u09c1 <span class=\"katex-eq\" data-katex-display=\"false\"> APM_1 <\/span><\/span><span style=\"font-weight: 400;\"> \u098f\u09ac\u0982 <span class=\"katex-eq\" data-katex-display=\"false\"> APM_2 <\/span><\/span><span style=\"font-weight: 400;\"> \u09b8\u09ae\u0995\u09cb\u09a3\u09c0 \u09a4\u09cd\u09b0\u09bf\u09ad\u09c1\u099c\u09a6\u09cd\u09ac\u09af\u09bc \u09b8\u09b0\u09cd\u09ac\u09b8\u09ae\u0964 \u0985\u09a4\u098f\u09ac, <span class=\"katex-eq\" data-katex-display=\"false\"> PM_1 <\/span><\/span><span style=\"font-weight: 400;\"> \u0993 <span class=\"katex-eq\" data-katex-display=\"false\"> PM_1 <\/span><\/span><span style=\"font-weight: 400;\">\u00a0 x \u09b2\u09ae\u09cd\u09ac\u09a6\u09cd\u09ac\u09af\u09bc\u09c7\u09b0 \u09a6\u09c8\u09b0\u09cd\u0998\u09cd\u09af \u09b8\u09ae\u09be\u09a8\u0964<\/span><\/p>\n<p><img loading=\"lazy\" class=\"aligncenter wp-image-3650 size-large\" src=\"https:\/\/10minuteschool.com\/content\/wp-content\/uploads\/2021\/12\/\u09b8\u09b0\u09b2-\u09b0\u09c7\u0996\u09be-1024x960.png\" alt=\"\u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995 \u0995\u09be\u0995\u09c7 \u09ac\u09b2\u09c7\" width=\"1024\" height=\"960\" srcset=\"https:\/\/10minuteschool.com\/content\/wp-content\/uploads\/2021\/12\/\u09b8\u09b0\u09b2-\u09b0\u09c7\u0996\u09be-1024x960.png 1024w, https:\/\/10minuteschool.com\/content\/wp-content\/uploads\/2021\/12\/\u09b8\u09b0\u09b2-\u09b0\u09c7\u0996\u09be-300x281.png 300w, https:\/\/10minuteschool.com\/content\/wp-content\/uploads\/2021\/12\/\u09b8\u09b0\u09b2-\u09b0\u09c7\u0996\u09be-768x720.png 768w, https:\/\/10minuteschool.com\/content\/wp-content\/uploads\/2021\/12\/\u09b8\u09b0\u09b2-\u09b0\u09c7\u0996\u09be.png 1052w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/p>\n<p><span style=\"font-weight: 400;\">\u0985\u09b0\u09cd\u09a5\u09be\u09ce, <span class=\"katex-eq\" data-katex-display=\"false\"> \\frac{a_{1} x_{1}+b_{1} y_{1}+c_{1}}{\\sqrt{a_{1}{ }^{2}+b_{1}{ }^{2}}} = \\frac{a_{2} x_{1}+b_{2} y_{1}+c_{2}}{\\sqrt{a_{2}{ }^{2}+b_{2}{ }^{2}}} \\quad \\cdots \\quad \\cdots \\quad \\cdots\u00a0 \\text{(iii)}<\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u0986\u09ac\u09be\u09b0 \u09af\u09c7\u09b9\u09c7\u09a4\u09c1 \u09ae\u09c2\u09b2\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u098f\u09ac\u0982 P \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 AL \u098f\u09ac\u0982 AR \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09a6\u09cd\u09ac\u09af\u09bc\u09c7\u09b0 \u098f\u0995\u0987 \u09aa\u09be\u09b0\u09cd\u09b6\u09cd\u09ac\u09c7 \u0985\u09ac\u09b8\u09cd\u09a5\u09bf\u09a4\u0964 \u09b8\u09c1\u09a4\u09b0\u09be\u0982 <span class=\"katex-eq\" data-katex-display=\"false\"> PM_1 <\/span><\/span><span style=\"font-weight: 400;\"> \u098f\u09ac\u0982 <span class=\"katex-eq\" data-katex-display=\"false\"> PM_2 <\/span><\/span><span style=\"font-weight: 400;\">\u00a0\u09b2\u09ae\u09cd\u09ac\u09a6\u09cd\u09ac\u09af\u09bc\u09c7\u09b0 \u098f\u0995\u0987 \u099a\u09bf\u09b9\u09cd\u09a8 \u09b9\u09ac\u09c7\u0964<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u0985\u09a8\u09c1\u09b0\u09c2\u09aa\u09ad\u09be\u09ac\u09c7 <\/span><span style=\"font-weight: 400;\">MAR \u0995\u09cb\u09a3\u09c7\u09b0 \u09b8\u09ae\u09a6\u09cd\u09ac\u09bf\u0996\u09a3\u09cd\u09a1\u0995 AE \u09b0\u09c7\u0996\u09be\u09b0 \u0989\u09aa\u09b0 <span class=\"katex-eq\" data-katex-display=\"false\"> Q(x_1, y_2) <\/span><\/span><span style=\"font-weight: 400;\">\u00a0\u09af\u09c7 \u0995\u09cb\u09a8 \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09b9\u09b2\u09c7,<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u0986\u09ae\u09b0\u09be \u09aa\u09be\u0987, <span class=\"katex-eq\" data-katex-display=\"false\"> \\frac{\\mathrm{a}_{1} \\mathrm{x}_{1}+\\mathrm{b}_{1} \\mathrm{y}_{1}+\\mathrm{c}_{1}}{\\sqrt{\\mathrm{a}_{1}{ }^{2}+\\mathrm{b}_{1}{ }^{2}}}=\\frac{\\mathrm{a}_{2} \\mathrm{x}_{1}+\\mathrm{b}_{2} \\mathrm{y}_{1}+\\mathrm{c}_{2}}{\\sqrt{\\mathrm{a}_{2}{ }^{2}+\\mathrm{b}_{2}{ }^{2}}} \\quad \\ldots \\quad \\ldots \\quad \\ldots \\text { (iv) } <\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u098f\u0996\u09be\u09a8\u09c7, <span class=\"katex-eq\" data-katex-display=\"false\"> QN_1 <\/span><\/span><span style=\"font-weight: 400;\"> \u098f\u09ac\u0982 <span class=\"katex-eq\" data-katex-display=\"false\"> QN_2 <\/span><\/span><span style=\"font-weight: 400;\">\u00a0\u09b2\u09ae\u09cd\u09ac\u09a6\u09cd\u09ac\u09af\u09bc\u09c7\u09b0 \u099a\u09bf\u09b9\u09cd\u09a8 \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u09b9\u09ac\u09c7\u0964 \u0995\u09be\u09b0\u09a3, \u09ae\u09c2\u09b2\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u0993 Q \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 LM \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u098f\u0995\u0987 \u09aa\u09be\u09b0\u09cd\u09b6\u09c7, \u0995\u09bf\u09a8\u09cd\u09a4\u09c1 RS \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u09aa\u09be\u09b0\u09cd\u09b6\u09cd\u09ac\u09c7\u0964<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u0985\u09a4\u098f\u09ac, \u09b8\u09ae\u09a6\u09cd\u09ac\u09bf\u0996\u09a3\u09cd\u09a1\u0995\u09a6\u09cd\u09ac\u09af\u09bc\u09c7\u09b0 \u0989\u09aa\u09b0 \u09af\u09c7 \u0995\u09cb\u09a8\u09cb \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09b0 \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995 \u09a6\u09cd\u09ac\u09be\u09b0\u09be,<\/span><\/p>\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\frac{\\mathrm{a}_{1} \\mathrm{x}_{1}+\\mathrm{b}_{1} \\mathrm{y}_{1}+\\mathrm{c}_{1}}{\\sqrt{\\mathrm{a}_{1}{ }^{2}+\\mathrm{b}_{1}{ }^{2}}} = \\pm \\frac{\\mathrm{a}_{2} \\mathrm{x}_{1}+\\mathrm{b}_{2} \\mathrm{y}_{1}+\\mathrm{c}_{2}}{\\sqrt{\\mathrm{a}_{2}{ }^{2}+\\mathrm{b}_{2}{ }^{2}}} <\/span> \u00a0\u098f\u09b0 \u098f\u0995\u099f\u09bf \u09ac\u09be \u0985\u09aa\u09b0\u099f\u09bf \u09b8\u09bf\u09a6\u09cd\u09a7 \u09b9\u09af\u09bc\u0964<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09af\u09c7\u09b9\u09c7\u09a4\u09c1 \u09b8\u09ae\u09a6\u09cd\u09ac\u09bf\u0996\u09a3\u09cd\u09a1\u0995\u09a6\u09cd\u09ac\u09af\u09bc \u098f\u09ae\u09a8 \u0995\u09a4\u0995\u0997\u09c1\u09b2\u09cb \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09b0 \u09b8\u099e\u09cd\u099a\u09be\u09b0\u09aa\u09a5 \u09af\u09be \u09a5\u09c7\u0995\u09c7 \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09a6\u09cd\u09ac\u09af\u09bc\u09c7\u09b0 \u0989\u09aa\u09b0 \u0985\u0999\u09cd\u0995\u09bf\u09a4 \u09b2\u09ae\u09cd\u09ac\u09a6\u09cd\u09ac\u09af\u09bc\u09c7\u09b0 \u09a6\u09c8\u09b0\u09cd\u0998\u09cd\u09af \u09b8\u09ae\u09be\u09a8\u0964 \u0985\u09a4\u098f\u09ac, \u09a8\u09bf\u09b0\u09cd\u09a3\u09c7\u09af\u09bc \u09b8\u09ae\u09a6\u09cd\u09ac\u09bf\u0996\u09a3\u09cd\u09a1\u0995\u09a6\u09cd\u09ac\u09af\u09bc P \u098f\u09ac\u0982 Q \u098f\u09b0 \u09b8\u099e\u09cd\u099a\u09be\u09b0\u09aa\u09a5\u0964 \u09b8\u09c1\u09a4\u09b0\u09be\u0982 \u09a4\u09be\u09a6\u09c7\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3,<\/span><\/p>\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\frac{\\mathrm{a}_{1} \\mathrm{x}_{1}+\\mathrm{b}_{1} \\mathrm{y}_{1}+\\mathrm{c}_{1}}{\\sqrt{\\mathrm{a}_{1}{ }^{2}+\\mathrm{b}_{1}{ }^{2}}} = \\pm \\frac{\\mathrm{a}_{2} \\mathrm{x}_{1}+\\mathrm{b}_{2} \\mathrm{y}_{1}+\\mathrm{c}_{2}}{\\sqrt{\\mathrm{a}_{2}{ }^{2}+\\mathrm{b}_{2}{ }^{2}}} \\quad \\ldots \\quad \\ldots \\quad \\ldots \\text { (v) } <\/span><\/span><\/p>\n<p>&nbsp;<\/p>\n<p><b>\u09a8\u09cb\u099f-\u09e7<\/b><\/p>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">\u09af\u09a6\u09bf <span class=\"katex-eq\" data-katex-display=\"false\"> c_1, c_2 <\/span> <\/span><span style=\"font-weight: 400;\">\u098f\u0995\u0987 \u099a\u09bf\u09b9\u09cd\u09a8\u09ac\u09bf\u09b6\u09bf\u09b7\u09cd\u099f \u09b9\u09af\u09bc \u0985\u09b0\u09cd\u09a5\u09be\u09ce \u0989\u09ad\u09af\u09bc\u0987 (+) \u0985\u09a5\u09ac\u09be \u0989\u09ad\u09af\u09bc\u0987 (-) \u099a\u09bf\u09b9\u09cd\u09a8\u09af\u09c1\u0995\u09cd\u09a4 \u09b9\u09b2\u09c7, \u09a4\u09ac\u09c7 (+) \u099a\u09bf\u09b9\u09cd\u09a8\u09af\u09c1\u0995\u09cd\u09a4 \u09b8\u09ae\u09c0\u0995\u09a3\u099f\u09bf \u09ae\u09c2\u09b2\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a7\u09be\u09b0\u09c0 \u0995\u09cb\u09a3\u09c7\u09b0 \u09b8\u09ae\u09a6\u09cd\u09ac\u09bf\u0996\u09a3\u09cd\u09a1\u0995 \u09ac\u09c1\u099d\u09be\u09af\u09bc\u0964 (-) \u099a\u09bf\u09b9\u09cd\u09a8\u09af\u09c1\u0995\u09cd\u09a4 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u099f\u09bf \u0985\u09a8\u09cd\u09af \u09b8\u09ae\u09a6\u09cd\u09ac\u09bf\u0996\u09a3\u09cd\u09a1\u0995\u09c7\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u09a8\u09bf\u09b0\u09cd\u09a6\u09c7\u09b6 \u0995\u09b0\u09c7\u0964<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">\u09af\u09c7 \u09b8\u09ae\u09a6\u09cd\u09ac\u09bf\u0996\u09a3\u09cd\u09a1\u0995, \u09aa\u09cd\u09b0\u09a6\u09a4\u09cd\u09a4 \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09a6\u09cd\u09ac\u09af\u09bc\u09c7\u09b0 \u0985\u09a8\u09cd\u09a4\u09b0\u09cd\u09ad\u09c1\u0995\u09cd\u09a4 \u09b8\u09c2\u0995\u09cd\u09b7\u09cd\u09ae\u0995\u09cb\u09a3\u099f\u09bf\u0995\u09c7 \u09b8\u09ae\u09a6\u09cd\u09ac\u09bf\u0996\u09a3\u09cd\u09a1\u09bf\u09a4 \u0995\u09b0\u09c7, \u09a4\u09be \u09aa\u09cd\u09b0\u09a6\u09a4\u09cd\u09a4 \u09af\u09c7 \u0995\u09cb\u09a8\u09cb \u098f\u0995\u099f\u09bf \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09be\u09a5\u09c7 45<\/span><span style=\"font-weight: 400;\">\u00b0<\/span><span style=\"font-weight: 400;\"> \u0995\u09cb\u09a3\u09c7\u09b0 \u099a\u09c7\u09af\u09bc\u09c7 \u099b\u09cb\u099f \u0995\u09cb\u09a3 \u0989\u09ce\u09aa\u09a8\u09cd\u09a8 \u0995\u09b0\u09c7\u0964 \u09b8\u09c1\u09a4\u09b0\u09be\u0982 \u09af\u09a6\u09bf \u09a6\u09c7\u0996\u09be \u09af\u09be\u09af\u09bc \u09af\u09c7 \u098f\u0995\u099f\u09bf \u09b8\u09ae\u09a6\u09cd\u09ac\u09bf\u0996\u09a3\u09cd\u09a1\u0995 \u0993 \u09af\u09c7 \u0995\u09cb\u09a8\u09cb \u098f\u0995\u099f\u09bf \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u0985\u09a8\u09cd\u09a4\u09b0\u09cd\u09ad\u09c1\u0995\u09cd\u09a4 \u0995\u09cb\u09a3\u09c7\u09b0 \u099f\u09c7\u09a8\u099c\u09c7\u09a8\u09cd\u099f 1 \u098f\u09b0 \u099a\u09c7\u09af\u09bc\u09c7 \u0995\u09ae, \u09a4\u09ac\u09c7 \u0990 \u09b8\u09ae\u09a6\u09cd\u09ac\u09bf\u0996\u09a3\u09cd\u09a1\u0995 \u09aa\u09cd\u09b0\u09a6\u09a4\u09cd\u09a4 \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09a6\u09cd\u09ac\u09af\u09bc\u09c7\u09b0 \u0985\u09a8\u09cd\u09a4\u09b0\u09cd\u09ad\u09c1\u0995\u09cd\u09a4 \u09b8\u09c2\u0995\u09cd\u09b7\u09cd\u09ae\u0995\u09cb\u09a3\u099f\u09bf\u0995\u09c7 \u09b8\u09ae\u09a6\u09cd\u09ac\u09bf\u0996\u09a3\u09cd\u09a1\u09bf\u09a4 \u0995\u09b0\u09c7\u0964<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">\u0995\u09cb\u09a8\u09cb \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7 \u09af\u09a6\u09bf \u0995\u09cb\u09a8\u09cb \u0995\u09cb\u09a3\u09c7\u09b0 \u09ac\u09bf\u09b6\u09c7\u09b7 \u09b8\u09ae\u09a6\u09cd\u09ac\u09bf\u0996\u09a3\u09cd\u09a1\u0995\u09c7\u09b0 \u09aa\u09cd\u09b0\u09af\u09bc\u09cb\u099c\u09a8 \u09b9\u09af\u09bc, \u09a4\u09ac\u09c7 \u09b8\u09c7\u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7 \u09aa\u09cd\u09b0\u0995\u09c3\u09a4 \u099a\u09bf\u09a4\u09cd\u09b0 \u098f\u0995\u09c7\u0987 \u09a4\u09be \u0995\u09b0\u09a4\u09c7 \u09b9\u09ac\u09c7\u0964 \u099a\u09bf\u09a4\u09cd\u09b0\u09c7 \u09b8\u09ae\u09a6\u09cd\u09ac\u09bf\u0996\u09a3\u09cd\u09a1\u0995\u09c7\u09b0 \u09a2\u09be\u09b2 \u09b2\u0995\u09cd\u09b7\u09cd\u09af \u0995\u09b0\u09c7 \u099a\u09bf\u09b9\u09cd\u09a8\u09c7\u09b0 \u09a6\u09cd\u09ac\u09cd\u09af\u09b0\u09cd\u09a5\u09a4\u09be \u09a6\u09c2\u09b0\u09c0\u09ad\u09c2\u09a4 \u0995\u09b0\u09a4\u09c7 \u09b9\u09ac\u09c7\u0964<\/span><\/li>\n<\/ul>\n<p><b>\u09a8\u09cb\u099f-\u09e8<\/b><\/p>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">\u09ae\u09a7\u09cd\u09af\u09ac\u09b0\u09cd\u09a4\u09c0 \u0995\u09cb\u09a3\u09c7\u09b0 \u09b8\u09ae\u09a6\u09cd\u09ac\u09bf\u0996\u09a3\u09cd\u09a1\u0995\u09a6\u09cd\u09ac\u09af\u09bc \u09aa\u09b0\u09b8\u09cd\u09aa\u09b0 \u09b2\u09ae\u09cd\u09ac \u09b9\u09ac\u09c7\u0964<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">\u098f\u0995\u099f\u09bf \u09a6\u09cd\u09ac\u09bf\u0996\u09a3\u09cd\u09a1\u0995 \u0993 \u098f\u0995\u099f\u09bf \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u09ae\u09a7\u09cd\u09af\u09ac\u09b0\u09cd\u09a4\u09c0 \u0995\u09cb\u09a3\u09c7\u09b0 \u09a2\u09be\u09b2 1 \u098f\u09b0 \u099a\u09c7\u09af\u09bc\u09c7 \u099b\u09cb\u099f \u09b9\u09b2\u09c7 \u0990 \u09a6\u09cd\u09ac\u09bf\u0996\u09a3\u09cd\u09a1\u0995\u099f\u09bf \u09aa\u09cd\u09b0\u09a6\u09a4\u09cd\u09a4 \u09b0\u09c7\u0996\u09be \u09a6\u09c1\u0987\u099f\u09bf\u09b0 \u09ae\u09a7\u09cd\u09af\u09ac\u09b0\u09cd\u09a4\u09c0 \u09b8\u09c2\u0995\u09cd\u09b7\u09cd\u09ae\u0995\u09cb\u09a3\u099f\u09bf\u0995\u09c7 \u09b8\u09ae\u09a6\u09cd\u09ac\u09bf\u0996\u09a3\u09cd\u09a1\u09bf\u09a4 \u0995\u09b0\u09c7\u0964<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\"> a_1 a_2 + b_1 b_2 &lt; 0 <\/span> \u09b9\u09b2\u09c7 \u098b\u09a3\u09be\u09a4\u09cd\u09ae\u0995 \u099a\u09bf\u09b9\u09cd\u09a8 \u09b9\u09a4\u09c7 \u09aa\u09cd\u09b0\u09be\u09aa\u09cd\u09a4 \u09a6\u09cd\u09ac\u09bf\u0996\u09a3\u09cd\u09a1\u0995\u099f\u09bf \u09b8\u09c2\u0995\u09cd\u09b7\u09cd\u09ae\u0995\u09cb\u09a3\u09c7\u09b0 \u09b8\u09ae\u09a6\u09cd\u09ac\u09bf\u0996\u09a3\u09cd\u09a1\u0995\u0964<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">\u09aa\u09cd\u09b0\u09a4\u09cd\u09af\u09c7\u0995\u099f\u09bf \u09b8\u09ae\u09a6\u09cd\u09ac\u09bf\u0996\u09a3\u09cd\u09a1\u0995\u09c7\u09b0 \u09b8\u09be\u09aa\u09c7\u0995\u09cd\u09b7\u09c7 \u09b0\u09c7\u0996\u09be\u09a6\u09cd\u09ac\u09af\u09bc \u098f\u0995\u09c7 \u0985\u09aa\u09b0\u09c7\u09b0 \u09aa\u09cd\u09b0\u09a4\u09bf\u099a\u09cd\u099b\u09ac\u09bf\u0964<\/span><\/li>\n<\/ul>\n<p><b>\u0985\u09a8\u09c1\u09b8\u09bf\u09a6\u09cd\u09a7\u09be\u09a8\u09cd\u09a4:<\/b><\/p>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\"> f(x, y) = a_1 x + b_1 y + c = 0 <\/span> <\/span><span style=\"font-weight: 400;\">\u098f\u09ac\u0982 <span class=\"katex-eq\" data-katex-display=\"false\"> g(x, y) = a_2 x + b_2 y + c_2 = 0 <\/span><\/span><span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">\u09b0\u09c7\u0996\u09be\u09a6\u09cd\u09ac\u09af\u09bc\u09c7\u09b0 \u0985\u09a8\u09cd\u09a4\u09b0\u09cd\u09ad\u09c1\u0995\u09cd\u09a4 \u0995\u09cb\u09a3\u09c7\u09b0 \u09b8\u09ae\u09a6\u09cd\u09ac\u09bf\u0996\u09a3\u09cd\u09a1\u0995\u09c7\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3, <span class=\"katex-eq\" data-katex-display=\"false\"> \\frac{\\mathrm{a}_{1} \\mathrm{x}_{1}+\\mathrm{b}_{1} \\mathrm{y}_{1}+\\mathrm{c}_{1}}{\\sqrt{\\mathrm{a}_{1}{ }^{2}+\\mathrm{b}_{1}{ }^{2}}} = \\pm \\frac{\\mathrm{a}_{2} \\mathrm{x}_{1}+\\mathrm{b}_{2} \\mathrm{y}_{1}+\\mathrm{c}_{2}}{\\sqrt{\\mathrm{a}_{2}{ }^{2}+\\mathrm{b}_{2}{ }^{2}}} <\/span>\u00a0 \u00a0<\/span><\/li>\n<li><span style=\"font-weight: 400;\">\u00a0<span class=\"katex-eq\" data-katex-display=\"false\"> P(\\alpha, \\beta) <\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a7\u09be\u09b0\u09a3\u0995\u09be\u09b0\u09c0 \u0995\u09cb\u09a3\u099f\u09bf\u09b0 \u09b8\u09ae\u09a6\u09bf\u0996\u09a8\u09cd\u09a1\u0995 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u2018+\u2019 \u09b9\u09ac\u09c7 \u09af\u0996\u09a8 <span class=\"katex-eq\" data-katex-display=\"false\"> f(\\alpha, \\beta) \\times g(\\alpha, \\beta) <\/span> <\/span><span style=\"font-weight: 400;\">&gt;<\/span><span style=\"font-weight: 400;\"> 0 \u2018-\u2019 \u09b9\u09ac\u09c7,\u09af\u0996\u09a8 <span class=\"katex-eq\" data-katex-display=\"false\"> f(\\alpha, \\beta) \\times g(\\alpha, \\beta) <\/span> <\/span>\u00a0<span style=\"font-weight: 400;\">&lt;<\/span><span style=\"font-weight: 400;\"> 0\u00a0<\/span><\/li>\n<li><span style=\"font-weight: 400;\"> \u09ae\u09c2\u09b2\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09a7\u09be\u09b0\u09a3\u0995\u09be\u09b0\u09c0 \u0995\u09cb\u09a3\u099f\u09bf\u09b0 \u09b8\u09ae\u09a6\u09cd\u09ac\u09bf\u0996\u09a3\u09cd\u09a1\u0995\u09c7\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 &#8216;+&#8217; \u0985\u09a5\u09ac\u09be \u2018-&#8216; \u09b9\u09ac\u09c7 \u09af\u0996\u09a8 \u09af\u09a5\u09be\u0995\u09cd\u09b0\u09ae\u09c7 <span class=\"katex-eq\" data-katex-display=\"false\">c_1 \\times c_2 &gt; 0<\/span> \u0993 <span class=\"katex-eq\" data-katex-display=\"false\">c_1 \\times c_2 &lt; 0<\/span>,<\/span><\/li>\n<\/ul>\n<p><span style=\"font-weight: 400;\">iii. <span class=\"katex-eq\" data-katex-display=\"false\"> P(x^{\\prime}, y^{\\prime}) <\/span> <\/span><span style=\"font-weight: 400;\">\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u099f\u09bf \u09b0\u09c7\u0996\u09be\u09a6\u09cd\u09ac\u09af\u09bc\u09c7\u09b0 \u0985\u09a8\u09cd\u09a4\u09b0\u09cd\u09ad\u09c1\u0995\u09cd\u09a4 \u09b8\u09cd\u09a5\u09c2\u09b2\u0995\u09cb\u09a3\u09c7 \u0985\u09a5\u09ac\u09be \u09b8\u09c2\u0995\u09cd\u09b7\u09cd\u09ae\u0995\u09cb\u09a3\u09c7 \u0985\u09ac\u09b8\u09cd\u09a5\u09bf\u09a4 \u09b9\u09ac\u09c7, \u09af\u0996\u09a8 \u09af\u09a5\u09be\u0995\u09cd\u09b0\u09ae\u09c7 <span class=\"katex-eq\" data-katex-display=\"false\"> f(x^{\\prime}, y^{\\prime}) \\times g(x^{\\prime}, y^{\\prime})\u00a0 (a_1 a_2 + b_1 b_2) &gt; 0 \\text{ \u09ac\u09be }, &lt; 0 \u00a0<\/span><\/span> <span style=\"font-weight: 400;\">\u00a0<\/span><\/p>\n<p>2. <span class=\"katex-eq\" data-katex-display=\"false\"> \\mathrm{L}_{1}=\\mathrm{a}_{1} \\mathrm{x}+\\mathrm{b}_{1} \\mathrm{y}+\\mathrm{c}_{1}=0 \\text { \u098f\u09ac\u0982 } \\mathrm{L}_{2}=\\mathrm{a}_{2} \\mathrm{x}+\\mathrm{b}_{2} \\mathrm{y}+\\mathrm{c}_{2}=0 \\text { \u09b9\u09af\u09bc, \u09a4\u09ac\u09c7 } <\/span><\/p>\n<p><span style=\"font-weight: 400;\">i. <span class=\"katex-eq\" data-katex-display=\"false\"> \\mathrm{a}_{1} \\mathrm{a}_{2}+\\mathrm{b}_{1} \\mathrm{~b}_{2}&gt;0 \\text { \u09b9\u09b2\u09c7, } \\mathrm{L}_{1} \\text { \u0993 } \\mathrm{L}_{2} <\/span> \u098f\u09b0 \u09ae\u09a7\u09cd\u09af\u09ac\u09b0\u09cd\u09a4\u09c0 \u09b8\u09cd\u09a5\u09c2\u09b2\u0995\u09cb\u09a3\u09c7\u09b0 \u09b8\u09ae\u09a6\u09cd\u09ac\u09bf\u0996\u09a8\u09cd\u09a1\u0995 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3,<\/span><\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\frac{a_1 x+b_1 y+c_1}{\\sqrt{a_1^{2}+b_1^{2}}} = +\\frac{a_2 x + b_2 y + c_2}{\\sqrt{a_2^{2}+b_2^{2}}} <\/span>\n<p><span style=\"font-weight: 400;\">ii. <span class=\"katex-eq\" data-katex-display=\"false\"> \\mathrm{a}_{1} \\mathrm{a}_{2}+\\mathrm{b}_{1} \\mathrm{~b}_{2}&gt;0 \\text { \u09b9\u09b2\u09c7, } \\mathrm{L}_{1} \\text { \u0993 } \\mathrm{L}_{2} <\/span> \u098f\u09b0 \u09ae\u09a7\u09cd\u09af\u09ac\u09b0\u09cd\u09a4\u09c0 \u09b8\u09c2\u0995\u09cd\u09b7\u09cd\u09ae\u0995\u09cb\u09a3\u09c7\u09b0 \u09b8\u09ae\u09a6\u09cd\u09ac\u09bf\u0996\u09a8\u09cd\u09a1\u0995 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3,<\/span><\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\frac{a_1 x+b_1 y+c_1}{\\sqrt{a_1^{2}+b_1^{2}}} = -\\frac{a_2 x + b_2 y + c_2}{\\sqrt{a_2^{2}+b_2^{2}}} <\/span>\n<p><span style=\"font-weight: 400;\">iii. <span class=\"katex-eq\" data-katex-display=\"false\"> \\mathrm{a}_{1} \\mathrm{a}_{2}+\\mathrm{b}_{1} \\mathrm{~b}_{2} &lt; 0 \\text { \u09b9\u09b2\u09c7, } \\mathrm{L}_{1} \\text { \u0993 } \\mathrm{L}_{2} <\/span> <\/span><span style=\"font-weight: 400;\">\u00a0\u098f\u09b0 \u09ae\u09a7\u09cd\u09af\u09ac\u09b0\u09cd\u09a4\u09c0 \u09b8\u09cd\u09a5\u09c2\u09b2\u0995\u09cb\u09a3\u09c7\u09b0 \u09b8\u09ae\u09a6\u09cd\u09ac\u09bf\u0996\u09a8\u09cd\u09a1\u0995 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3,<\/span><\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\frac{a_1 x+b_1 y+c_1}{\\sqrt{a_1^{2}+b_1^{2}}} = -\\frac{a_2 x + b_2 y + c_2}{\\sqrt{a_2^{2}+b_2^{2}}} <\/span>\n<p><span style=\"font-weight: 400;\">iv. <span class=\"katex-eq\" data-katex-display=\"false\"> \\mathrm{a}_{1} \\mathrm{a}_{2}+\\mathrm{b}_{1} \\mathrm{~b}_{2} &lt; 0 \\text { \u09b9\u09b2\u09c7, } \\mathrm{L}_{1} \\text { \u0993 } \\mathrm{L}_{2} <\/span> <\/span><span style=\"font-weight: 400;\">\u00a0\u098f\u09b0 \u09ae\u09a7\u09cd\u09af\u09ac\u09b0\u09cd\u09a4\u09c0 \u09b8\u09c2\u0995\u09cd\u09b7\u09cd\u09ae\u0995\u09cb\u09a3\u09c7\u09b0 \u09b8\u09ae\u09a6\u09cd\u09ac\u09bf\u0996\u09a8\u09cd\u09a1\u0995 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3,<\/span><\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\frac{a_1 x+b_1 y+c_1}{\\sqrt{a_1^{2}+b_1^{2}}} = +\\frac{a_2 x + b_2 y + c_2}{\\sqrt{a_2^{2}+b_2^{2}}} <\/span>\n<p>&nbsp;<\/p>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><b>\u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u098f\u0995\u0987 \u0985\u09a5\u09ac\u09be \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u09aa\u09be\u09b0\u09cd\u09b6\u09cd\u09ac\u09c7 \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09b0 \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09a8 (<\/b><b>Position of the Points on the same side or opposite side of a straight line)<\/b><\/li>\n<\/ul>\n<p><span style=\"font-weight: 400;\">\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, MN \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 <\/span><span style=\"font-weight: 400;\">ax\u00a0+\u00a0by\u00a0+\u00a0c\u00a0=\u00a00.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09a7\u09b0\u09bf, <span class=\"katex-eq\" data-katex-display=\"false\"> P(x_1, y_1) <\/span><\/span><span style=\"font-weight: 400;\"> \u098f\u09ac\u0982 <span class=\"katex-eq\" data-katex-display=\"false\"> Q(x_2, y_2) <\/span><\/span><span style=\"font-weight: 400;\">\u00a0\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a6\u09cd\u09ac\u09af\u09bc\u09c7\u09b0 \u09b8\u0982\u09af\u09cb\u099c\u0995 \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be MN \u09b0\u09c7\u0996\u09be\u0995\u09c7 R(x, y) \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 \u099b\u09c7\u09a6 \u0995\u09b0\u09c7\u0964<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09a7\u09b0\u09bf, <\/span><span style=\"font-weight: 400;\">PR\u00a0:\u00a0RQ\u00a0=\u00a0k\u00a0:\u00a01<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\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{k x_{2}+x_{1}}{k+1}, \\frac{k y_{2}+y_{1}}{k+1}\\right) <\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09af\u09c7\u09b9\u09c7\u09a4\u09c1 R \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u099f\u09bf MN \u09b0\u09c7\u0996\u09be\u09b0 \u0989\u09aa\u09b0 \u0985\u09ac\u09b8\u09cd\u09a5\u09bf\u09a4\u0964<\/span><\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\"> \\mathrm{a} \\frac{k x_{2}+x_{1}}{k+1}+\\mathrm{b} \\frac{k x_{2}+y_{1}}{k+1}+\\mathrm{c}=0 <\/span>\n<p><span style=\"font-weight: 400;\">\u09ac\u09be, <span class=\"katex-eq\" data-katex-display=\"false\"> a\\left(k x_{2}+x_{1}\\right)+b\\left(y_{2}+y_{1}\\right)+c(k+1)=0 <\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09ac\u09be, <span class=\"katex-eq\" data-katex-display=\"false\"> \\mathrm{k}\\left(\\mathrm{ax}_{2}+\\mathrm{by}_{2}+\\mathrm{c}\\right)+\\mathrm{ax}_{1}+\\mathrm{bx}_{1}+\\mathrm{c}=0 <\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09ac\u09be, <span class=\"katex-eq\" data-katex-display=\"false\"> \\mathrm{k}=-\\frac{a x_{1}+b y_{1}+c}{a x_{2}+b y_{2}+c} <\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">P \u0993 Q \u09af\u09a6\u09bf MN \u09b0\u09c7\u0996\u09be\u09b0 \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u09aa\u09be\u09b0\u09cd\u09b6\u09cd\u09ac\u09c7 \u0985\u09ac\u09b8\u09cd\u09a5\u09bf\u09a4 \u09b9\u09af\u09bc, \u09a4\u09be\u09b9\u09b2\u09c7 R, PQ \u09b0\u09c7\u0996\u09be\u0982\u09b6\u09c7\u09b0 \u0985\u09a8\u09cd\u09a4\u0983\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09b9\u09ac\u09c7\u0964<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09b8\u09c1\u09a4\u09b0\u09be\u0982 k : 1 \u0985\u09a8\u09c1\u09aa\u09be\u09a4\u099f\u09bf \u09a7\u09a8\u09be\u09a4\u09cd\u09ae\u0995 \u09b9\u09ac\u09c7\u0964 \u0986\u09ac\u09be\u09b0, P \u0993 Q \u09af\u09a6\u09bf MN \u09b0\u09c7\u0996\u09be\u09b0 \u098f\u0995\u0987 \u09aa\u09be\u09b0\u09cd\u09b6\u09cd\u09ac\u09c7 \u0985\u09ac\u09b8\u09cd\u09a5\u09bf\u09a4 \u09b9\u09af\u09bc, \u09a4\u09be\u09b9\u09b2\u09c7 R, PQ \u09b0\u09c7\u0996\u09be\u0982\u09b6\u09c7\u09b0 \u09ac\u09b9\u09bf\u0983\u09b8\u09cd\u09a5 \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09b9\u09ac\u09c7\u0964 \u09b8\u09c1\u09a4\u09b0\u09be\u0982 k : 1 \u0985\u09a8\u09c1\u09aa\u09be\u09a4\u099f\u09bf \u098b\u09a3\u09be\u09a4\u09cd\u09ae\u0995 \u09b9\u09ac\u09c7\u0964<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09b8\u09c1\u09a4\u09b0\u09be\u0982 <span class=\"katex-eq\" data-katex-display=\"false\"> \\frac{a x_{1}+b y_{1}+c}{a x_{2}+b y_{2}+c} <\/span><\/span><span style=\"font-weight: 400;\"> \u098b\u09a3\u09be\u09a4\u09cd\u09ae\u0995 \u0985\u09a5\u09ac\u09be \u09a7\u09a8\u09be\u09a4\u09cd\u09ae\u0995 \u09b9\u09b2\u09c7 \u0985\u09b0\u09cd\u09a5\u09be\u09ce <span class=\"katex-eq\" data-katex-display=\"false\"> a x_{1}+b y_{1}+c <\/span><\/span><span style=\"font-weight: 400;\"> \u098f\u09ac\u0982 <span class=\"katex-eq\" data-katex-display=\"false\"> a x_{2}+b y_{2}+c <\/span><\/span><span style=\"font-weight: 400;\">\u00a0\u098f\u0995\u0987 \u09ac\u09be \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u099a\u09bf\u09b9\u09cd\u09a8\u09af\u09c1\u0995\u09cd\u09a4 \u09b9\u09b2\u09c7, P \u0993 Q \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a6\u09cd\u09ac\u09af\u09bc MN \u09b0\u09c7\u0996\u09be\u09b0 \u098f\u0995\u0987 \u09ac\u09be \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u09aa\u09be\u09b0\u09cd\u09b6\u09cd\u09ac\u09c7 \u0985\u09ac\u09b8\u09cd\u09a5\u09bf\u09a4 \u09b9\u09ac\u09c7\u0964<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09a7\u09a8\u09be\u09a4\u09cd\u09ae\u0995 \u0993 \u098b\u09a3\u09be\u09a4\u09cd\u09ae\u0995 \u09aa\u09be\u09b0\u09cd\u09b6\u09cd\u09ac (Positive and negative side): <\/span><span style=\"font-weight: 400;\">ax\u00a0+\u00a0by\u00a0+\u00a0c\u00a0=\u00a00\u00a0<\/span><span style=\"font-weight: 400;\">\u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u0995\u09cb\u09a8\u09cb \u09aa\u09be\u09b0\u09cd\u09b6\u09cd\u09ac\u09c7\u09b0 \u09af\u09c7 \u0995\u09cb\u09a8\u09cb \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;\">\u00a0\u098f\u09b0 \u099c\u09a8\u09cd\u09af \u09af\u09a6\u09bf <span class=\"katex-eq\" data-katex-display=\"false\"> a x_{1}+b y_{1}+c <\/span> <\/span><span style=\"font-weight: 400;\">\u00a0\u09b8\u09b0\u09cd\u09ac\u09a6\u09be \u09a7\u09a8\u09be\u09a4\u09cd\u09ae\u0995 \u09b9\u09af\u09bc \u09a4\u09ac\u09c7 \u0990 \u09aa\u09be\u09b0\u09cd\u09b6\u099f\u09bf\u0995\u09c7 \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u099f\u09bf\u09b0 \u09a7\u09a8\u09be\u09a4\u09cd\u09ae\u0995 \u09aa\u09be\u09b0\u09cd\u09b6\u09cd\u09ac \u098f\u09ac\u0982 \u09a4\u09be\u09b0 \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u09aa\u09be\u09b0\u09cd\u09b6\u09cd\u09ac\u099f\u09bf\u0995\u09c7 \u098b\u09a3\u09be\u09a4\u09cd\u09ae\u0995 \u09aa\u09be\u09b0\u09cd\u09b6\u09cd\u09ac \u09ac\u09b2\u09be \u09b9\u09af\u09bc\u0964<\/span><\/p>\n<p><b>\u09ae\u09c2\u09b2\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09b0 \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09a8<\/b><span style=\"font-weight: 400;\"> : <\/span><span style=\"font-weight: 400;\">\u09af\u09a6\u09bf <\/span><span style=\"font-weight: 400;\">ax\u00a0+\u00a0by\u00a0+\u00a0c\u00a0=\u00a00\u00a0<\/span><span style=\"font-weight: 400;\">\u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09c7\u09b0 c \u09a7\u09a8\u09be\u09a4\u09cd\u09ae\u0995 \u09b9\u09af\u09bc, \u09a4\u09ac\u09c7 \u09ae\u09c2\u09b2\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 (0, 0) \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u09a7\u09a8\u09be\u09a4\u09cd\u09ae\u0995 \u09aa\u09be\u09b0\u09cd\u09b6\u09cd\u09ac\u09c7 \u098f\u09ac\u0982 c \u098b\u09a3\u09be\u09a4\u09cd\u09ae\u0995 \u09b9\u09b2\u09c7 \u09ae\u09c2\u09b2\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09b0\u09c7\u0996\u09be\u099f\u09bf\u09b0 \u098b\u09a3\u09be\u09a4\u09cd\u09ae\u0995 \u09aa\u09be\u09b0\u09cd\u09b6\u09cd\u09ac\u09c7 \u0985\u09ac\u09b8\u09cd\u09a5\u09bf\u09a4\u0964<\/span><\/p>\n<p><b>\u09ae\u09c2\u09b2\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u0993 \u0985\u09aa\u09b0 \u09af\u09c7 \u0995\u09cb\u09a8\u09cb \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09b0 \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09a8 : <\/b><span style=\"font-weight: 400;\">\u09af\u09a6\u09bf <span class=\"katex-eq\" data-katex-display=\"false\"> a x_{1}+b y_{1}+c <\/span> <\/span><span style=\"font-weight: 400;\">\u098f\u09ac\u0982 c \u098f\u0995\u0987 \u099a\u09bf\u09b9\u09cd\u09a8\u09ac\u09bf\u09b6\u09bf\u09b7\u09cd\u099f \u09b9\u09af\u09bc, \u09a4\u09ac\u09c7 (0, 0) \u098f\u09ac\u0982 <span class=\"katex-eq\" data-katex-display=\"false\"> (x_1, y_1) <\/span>\u00a0<\/span><span style=\"font-weight: 400;\"> \u098f\u0995\u0987 \u09aa\u09be\u09b0\u09cd\u09b6\u09cd\u09ac\u09c7 \u0985\u09ac\u09b8\u09cd\u09a5\u09bf\u09a4 \u09b9\u09ac\u09c7\u0964 \u0986\u09b0 \u09af\u09a6\u09bf \u09ad\u09bf\u09a8\u09cd\u09a8 \u099a\u09bf\u09b9\u09cd\u09a8\u09ac\u09bf\u09b6\u09bf\u09b7\u09cd\u099f \u09b9\u09af\u09bc \u09a4\u09ac\u09c7 \u09ae\u09c2\u09b2\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u0993 <span class=\"katex-eq\" data-katex-display=\"false\"> (x_1, y_1) <\/span>\u00a0<\/span><span style=\"font-weight: 400;\">\u00a0\u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u09aa\u09be\u09b0\u09cd\u09b6\u09cd\u09ac\u09c7 \u0985\u09ac\u09b8\u09cd\u09a5\u09bf\u09a4 \u09b9\u09ac\u09c7\u0964\u00a0<\/span><\/p>\n<p><b>\u0989\u09a6\u09be\u09b9\u09b0\u09a3: A(2, 5), B(-1, 3) \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a6\u09cd\u09ac\u09af\u09bc 3x -2y + 7 = 0 \u09b0\u09c7\u0996\u09be\u09b0 \u098f\u0995\u0987 \u09aa\u09be\u09b0\u09cd\u09b6\u09cd\u09ac\u09c7 \u0985\u09a5\u09ac\u09be \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u09aa\u09be\u09b0\u09cd\u09b6\u09cd\u09ac\u09c7 \u0985\u09ac\u09b8\u09cd\u09a5\u09bf\u09a4 \u0995\u09bf\u09a8\u09be \u09a4\u09be \u09a8\u09bf\u09b0\u09cd\u09a3\u09af\u09bc \u0995\u09b0\u0964 \u0995\u09cb\u09a8 \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u099f\u09bf \u09ae\u09c2\u09b2\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09b0 \u09aa\u09be\u09b0\u09cd\u09b6\u09cd\u09ac\u09c7 \u0985\u09ac\u09b8\u09cd\u09a5\u09bf\u09a4?<\/b><\/p>\n<p><b>\u09b8\u09ae\u09be\u09a7\u09be\u09a8 : <\/b><span style=\"font-weight: 400;\">\u09a7\u09b0\u09bf, \u09aa\u09cd\u09b0\u09a6\u09a4\u09cd\u09a4 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3, <\/span><span style=\"font-weight: 400;\">L\u00a0=\u00a03x\u00a0&#8211;\u00a02y\u00a0+7\u00a0=\u00a00<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u2234\u00a0L(A)\u00a0=\u00a03.2\u00a0\u2013\u00a02.5\u00a0+7\u00a0=\u00a03\u00a0&gt;\u00a00<\/span><b>\u00a0<\/b><\/p>\n<p><span style=\"font-weight: 400;\">\u098f\u09ac\u0982, <\/span><span style=\"font-weight: 400;\">L(B)\u00a0=\u00a03.(-1)\u00a0\u2013\u00a02.3\u00a0+7\u00a0=\u00a0-2\u00a0&lt;\u00a00<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09a6\u09c7\u0996\u09be\u0993 \u09af\u09c7, L(A) \u098f\u09ac\u0982 L(B) \u09aa\u09b0\u09b8\u09cd\u09aa\u09b0 \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u099a\u09bf\u09b9\u09cd\u09a8\u09af\u09c1\u0995\u09cd\u09a4\u0964 \u09b8\u09c1\u09a4\u09b0\u09be\u0982 \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a6\u09cd\u09ac\u09af\u09bc \u09aa\u09cd\u09b0\u09a6\u09a4\u09cd\u09a4 \u09b0\u09c7\u0996\u09be\u09b0 \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u09aa\u09be\u09b0\u09cd\u09b6\u09cd\u09ac\u09c7 \u0985\u09ac\u09b8\u09cd\u09a5\u09bf\u09a4\u0964 \u0986\u09ac\u09be\u09b0 \u09ae\u09c2\u09b2\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1, O(0, 0).<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u2234\u00a0L(O)\u00a0=\u00a03.0\u00a0\u2013\u00a02.0\u00a0+7\u00a0=\u00a07\u00a0&gt;\u00a00<\/span><b>\u00a0<\/b><\/p>\n<p><span style=\"font-weight: 400;\">\u09b8\u09c1\u09a4\u09b0\u09be\u0982 L(A) \u098f\u09ac\u0982 L(0) \u0989\u09ad\u09af\u09bc\u09c7\u0987 \u09a7\u09a8\u09be\u09a4\u09cd\u09ae\u0995 \u0985\u09b0\u09cd\u09a5\u09be\u09ce \u098f\u0995\u0987 \u099a\u09bf\u09b9\u09cd\u09a8\u09af\u09c1\u0995\u09cd\u09a4\u0964 \u0985\u09a4\u098f\u09ac \u09b0\u09c7\u0996\u09be\u099f\u09bf\u09b0 \u09af\u09c7 \u09aa\u09be\u09b0\u09cd\u09b6\u09cd\u09ac\u09c7 \u09ae\u09c2\u09b2\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u0990 \u09aa\u09be\u09b0\u09cd\u09b6\u09cd\u09ac\u09c7 A \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u099f\u09bf \u0985\u09ac\u09b8\u09cd\u09a5\u09bf\u09a4\u0964<\/span><\/p>\n<p>&nbsp;<\/p>\n<hr \/>\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<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 &#8211; \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 &#8211; \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 &#8211; \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 &#8211; \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 &#8211; \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","protected":false},"excerpt":{"rendered":"<p>\u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995 \u0995\u09be\u0995\u09c7 \u09ac\u09b2\u09c7? \u098f\u09ac\u0982 \u098f\u09b0 \u09ac\u09bf\u09b8\u09cd\u09a4\u09be\u09b0\u09bf\u09a4 \u09aa\u09cd\u09b0\u0995\u09be\u09b0\u09ad\u09c7\u09a6 \u09b8\u09ae\u09a4\u09b2\u09c7 \u09aa\u09cb\u09b2\u09be\u09b0 \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995 \u0993 \u0995\u09be\u09b0\u09cd\u09a4\u09c7\u09b8\u09c0\u09af\u09bc \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995 (Cartesian and Polar co-ordinates on plane): \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995 \u0995\u09be\u0995\u09c7 \u09ac\u09b2\u09c7? (Co-ordinates): \u0995\u09cb\u09a8 \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09ac\u09be \u09ac\u09b8\u09cd\u09a4\u09c1\u09b0 \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09a8 \u09b8\u09c1\u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f\u09ad\u09be\u09ac\u09c7 \u09a8\u09bf\u09b0\u09cd\u09a3\u09af\u09bc\u09c7\u09b0 \u099c\u09a8\u09cd\u09af \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995 \u09ac\u09cd\u09af\u09ac\u09b9\u09be\u09b0 \u0995\u09b0\u09be \u09b9\u09af\u09bc\u0964 \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995 \u09ac\u09cd\u09af\u09ac\u09b8\u09cd\u09a5\u09be \u09a6\u09c1&#8217;\u09a7\u09b0\u09a8\u09c7\u09b0\u0964 \u09af\u09a5\u09be- i)<\/p>\n<p> <a class=\"redmore\" href=\"https:\/\/10minuteschool.com\/content\/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":[2512,2516,2518,2515,2513,2511,2514,2517],"_links":{"self":[{"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/posts\/3645"}],"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=3645"}],"version-history":[{"count":16,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/posts\/3645\/revisions"}],"predecessor-version":[{"id":16093,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/posts\/3645\/revisions\/16093"}],"wp:attachment":[{"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/media?parent=3645"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/categories?post=3645"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/tags?post=3645"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}