{"id":1014,"date":"2024-01-30T12:17:29","date_gmt":"2024-01-30T06:17:29","guid":{"rendered":"https:\/\/stage-wp.10minuteschool.com\/?p=1014"},"modified":"2024-11-03T21:06:26","modified_gmt":"2024-11-03T15:06:26","slug":"equation-of-circle-in-polar-coordinates","status":"publish","type":"post","link":"https:\/\/10minuteschool.com\/content\/equation-of-circle-in-polar-coordinates\/","title":{"rendered":"\u09aa\u09cb\u09b2\u09be\u09b0 \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995\u09c7 \u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 (Equation of circle in polar coordinates)"},"content":{"rendered":"

\u09aa\u09cb\u09b2\u09be\u09b0 \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995\u09c7 \u0995\u09be\u0995\u09c7 \u09ac\u09b2\u09c7?<\/span><\/h2>\n

\u09aa\u09cb\u09b2\u09be\u09b0 \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995<\/strong> \u09b9\u09b2 \u098f\u0995\u099f\u09bf \u0997\u09be\u09a3\u09bf\u09a4\u09bf\u0995 \u09aa\u09a6\u09cd\u09a7\u09a4\u09bf \u09af\u09be\u09b0 \u09ae\u09be\u09a7\u09cd\u09af\u09ae\u09c7 \u0986\u09ae\u09b0\u09be \u0995\u09cb\u09a8\u09cb \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09b0 \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09a8 \u09a8\u09bf\u09b0\u09cd\u09a6\u09c7\u09b6 \u0995\u09b0\u09a4\u09c7 \u09aa\u09be\u09b0\u09bf\u0964 \u098f\u099f\u09bf \u09ae\u09c2\u09b2\u09a4 \u098f\u0995\u099f\u09bf \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 (\u09af\u09be\u0995\u09c7 \u09ae\u09c2\u09b2\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09ac\u09be \u09aa\u09cb\u09b2 \u09ac\u09b2\u09be \u09b9\u09af\u09bc) \u09a5\u09c7\u0995\u09c7 \u09b8\u09c7\u0987 \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09b0 \u09a6\u09c2\u09b0\u09a4\u09cd\u09ac \u098f\u09ac\u0982 \u098f\u0995\u099f\u09bf \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09b0\u09c7\u0996\u09be (\u09af\u09be\u0995\u09c7 \u0986\u09a6\u09bf\u09b0\u09c7\u0996\u09be \u09ac\u09be \u09aa\u09cb\u09b2\u09be\u09b0 \u0985\u0995\u09cd\u09b7 \u09ac\u09b2\u09be \u09b9\u09af\u09bc) \u09a5\u09c7\u0995\u09c7 \u0995\u09cb\u09a3 \u09a6\u09bf\u09af\u09bc\u09c7 \u09a8\u09bf\u09b0\u09cd\u09a7\u09be\u09b0\u09bf\u09a4 \u09b9\u09af\u09bc\u0964<\/p>\n

\"\u09aa\u09cb\u09b2\u09be\u09b0<\/p>\n

\u0995\u09be\u09b0\u09cd\u09a4\u09c7\u09b8\u09c0\u09df \u09b8\u09ae\u09a4\u09b2\u09c7 \u098f\u0995\u099f\u09bf \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 <\/span>P<\/span> \u098f\u09b0 \u0995\u09be\u09b0\u09cd\u09a4\u09c7\u09b8\u09c0\u09df \u0993 \u09aa\u09cb\u09b2\u09be\u09b0 \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995 \u09af\u09a5\u09be\u0995\u09cd\u09b0\u09ae\u09c7 (x,\u00a0y)<\/span><\/span> \u0993 (r,\u00a0\\theta)<\/span><\/span> \u09b9\u09b2\u09c7 x=r\\cos \\theta,\u00a0y=r \\sin \\theta \\; \u098f\u09ac\u0982\\; x^2+y^2=r^2<\/span><\/span>\u0964<\/span><\/p>\n

\u0986\u09ac\u09be\u09b0,<\/span> C(h,\u00a0k)<\/span> <\/span>\u0995\u09cb\u09a8\u09cb \u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u0995\u09c7\u09a8\u09cd\u09a6\u09cd\u09b0, a<\/span><\/span> \u09ac\u09cd\u09af\u09be\u09b8\u09be\u09b0\u09cd\u09a7 \u098f\u09ac\u0982 P(x,\u00a0y)<\/span> <\/span>\u09ac\u09c3\u09a4\u09cd\u09a4\u099f\u09bf\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 \u09b9\u09b2\u09c7 \u0989\u0995\u09cd\u09a4 \u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3,<\/span>\u00a0<\/span>\u00a0<\/span><\/p>\n(x-h)^2+(y-k)^2=a^2......(i)<\/span>\n

(h, k)<\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09b0 \u09aa\u09cb\u09b2\u09be\u09b0 \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995 (r_0,\u00a0\\phi)<\/span> <\/span>\u09b9\u09b2\u09c7, h=r_0 \\cos \\phi,\u00a0k=r_0 \\sin \\theta\u00a0<\/span><\/span><\/p>\n

(i)<\/span><\/span>\u00a0\u09a8\u0982 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u09b9\u09a4\u09c7 \u09aa\u09be\u0987,<\/span>\u00a0<\/span><\/p>\n(r \\cos \\theta-r_0 \\cos \\theta)^2+(r \\sin \\theta-r_0 \\sin \\phi)^2=a^2\\\\\n\nr^2(\\cos^2 \\theta + \\sin^2 \\theta)-2rr_0 (\\cos \\theta \\cos \\phi + \\sin \\theta \\sin \\phi)+r_0^2(\\cos^2 \\phi +\\sin^2 \\phi)=a^2\\\\\n\nr^2-2rr_0\\cos \\theta-\\phi+r_0^2=a^2 [\\because \\cos \\theta \\cos \\phi+\\sin \\theta \\sin \\phi=\\cos \\theta-\\phi]<\/span>\n

\u2234<\/span>\u09aa\u09cb\u09b2\u09be\u09b0 \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995\u09c7 \u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0<\/a><\/span> \u09b8\u09ae\u09c0\u0995\u09b0\u09a3, r^2-2rr_0\\cos(\\theta-\\phi)+r_0^2=a^2......(ii)\u00a0<\/span><\/span><\/p>\n

\u09af\u09c7\u0996\u09be\u09a8\u09c7, a<\/span><\/span> \u09b9\u09b2\u09cb \u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u09ac\u09cd\u09af\u09be\u09b8\u09be\u09b0\u09cd\u09a7, (r,\\theta)<\/span><\/span> \u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u0989\u09aa\u09b0 \u09af\u09c7 \u0995\u09cb\u09a8\u09cb \u09b8\u09be\u09a7\u09be\u09b0\u09a3 \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09aa\u09cb\u09b2\u09be\u09b0 \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995 \u098f\u09ac\u0982 (r_0,\\phi)<\/span><\/span>\u00a0\u09b9\u09b2\u09cb \u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u0995\u09c7\u09a8\u09cd\u09a6\u09cd\u09b0\u09c7\u09b0 \u09aa\u09cb\u09b2\u09be\u09b0 \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995\u0964<\/span><\/p>\n

\u0986\u09ac\u09be\u09b0,<\/span>\u00a0<\/span>\u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u09b8\u09be\u09a7\u09be\u09b0\u09a3 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09c7\u09b0 \u099c\u09a8\u09cd\u09af (ii)<\/span><\/span> \u09a8\u0982 \u09b9\u09a4\u09c7 \u09aa\u09be\u0987,\u00a0<\/span>r^2-2r(r_0\\cos \\theta \\cos \\phi+r_0 \\sin \\theta \\sin \\phi)+r_0^2-a^2=0\\\\\n\nr^2+2r{(-r_0 \\cos \\phi)\\cos \\theta+(-r_0 \\sin \\phi)\\sin \\theta}+r_0^2-a^2=0\\\\\n\nr^2+2r(g \\cos \\theta+f \\sin \\theta)+c=0......(iii)\u00a0<\/span><\/p>\n

\u09af\u09c7\u0996\u09be\u09a8\u09c7, g=-r_0 \\cos \\phi,\u00a0f=-r_0 \\sin \\phi,\u00a0c=r_0^2-a^2<\/span><\/span><\/p>\n

(iii)<\/span><\/span>\u00a0\u09a8\u0982 \u09b9\u09b2\u09cb \u09aa\u09cb\u09b2\u09be\u09b0 \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995\u09c7\u09b0 \u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u09b8\u09be\u09a7\u09be\u09b0\u09a3 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u0964<\/span><\/p>\n

\u098f\u0996\u09a8<\/span>, g^2+f^2=r_0^2(\\cos^2 \\phi+\\sin^2 \\phi)=r_0^2 \\therefore r_0=\\sqrt{g^2+f^2}\n\n\u098f\u09ac\u0982\u00a0\\frac{f}{g}=\\frac{-r_0 \\sin \\phi}{-r_0 \\cos \\phi}=\\tan \\phi \\therefore \\phi=\\tan^{-1}\\Big(\\frac{f}{g}\\Big)<\/span><\/span><\/p>\n

\u2234<\/span> \u0995\u09c7\u09a8\u09cd\u09a6\u09cd\u09b0\u09c7\u09b0 \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995, (r,\u00a0\\phi)=\\Big(\\sqrt{g^2+f^2},\\tan^{-1}\\Big(\\frac{f}{g}\\Big)\\Big)<\/span><\/span> \u098f\u09ac\u0982 \u09ac\u09cd\u09af\u09be\u09b8\u09be\u09b0\u09cd\u09a7 a=\\sqrt{r_0^2-c}=\\sqrt{g^2+f^2-c\u00a0}<\/span><\/span><\/p>\n

\u09ac\u09bf\u09b6\u09c7\u09b7 \u0985\u09ac\u09b8\u09cd\u09a5\u09be:<\/b><\/p>\n