{"id":1009,"date":"2024-01-30T12:19:04","date_gmt":"2024-01-30T06:19:04","guid":{"rendered":"https:\/\/stage-wp.10minuteschool.com\/?p=1009"},"modified":"2024-11-03T20:23:31","modified_gmt":"2024-11-03T14:23:31","slug":"equation-of-circle-with-fixed-center-and-radius","status":"publish","type":"post","link":"https:\/\/10minuteschool.com\/content\/equation-of-circle-with-fixed-center-and-radius\/","title":{"rendered":"\u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u0995\u09c7\u09a8\u09cd\u09a6\u09cd\u09b0 \u0993 \u09ac\u09cd\u09af\u09be\u09b8\u09be\u09b0\u09cd\u09a7\u09ac\u09bf\u09b6\u09bf\u09b7\u09cd\u099f \u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 (Equation of a circle with fixed center and radius)\u00a0"},"content":{"rendered":"\r\n

\u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u0995\u09c7\u09a8\u09cd\u09a6\u09cd\u09b0 \u098f\u09ac\u0982 \u09ac\u09cd\u09af\u09be\u09b8\u09be\u09b0\u09cd\u09a7 \u0995\u09be\u0995\u09c7 \u09ac\u09b2\u09c7?<\/span><\/h2>\r\n

\u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0<\/a><\/span> \u0995\u09c7\u09a8\u09cd\u09a6\u09cd\u09b0 \u09b9\u09b2 \u09b8\u09c7\u0987 \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09af\u09be \u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u09b8\u0995\u09b2 \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09a5\u09c7\u0995\u09c7 \u09b8\u09ae\u09be\u09a8 \u09a6\u09c2\u09b0\u09a4\u09cd\u09ac\u09c7 \u0985\u09ac\u09b8\u09cd\u09a5\u09bf\u09a4\u0964 \u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0<\/a><\/span> \u0995\u09c7\u09a8\u09cd\u09a6\u09cd\u09b0 \u09a5\u09c7\u0995\u09c7 \u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u09af\u09c7\u0995\u09cb\u09a8\u09cb \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09aa\u09b0\u09cd\u09af\u09a8\u09cd\u09a4 \u09af\u09c7 \u09b0\u09c7\u0996\u09be\u0982\u09b6, \u09a4\u09be\u0995\u09c7 \u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u09ac\u09cd\u09af\u09be\u09b8\u09be\u09b0\u09cd\u09a7 \u09ac\u09b2\u09c7\u0964<\/p>\r\n\r\n\r\n\r\n

\u0995\u09c7\u09a8\u09cd\u09a6\u09cd\u09b0 (h,k)<\/span> \u098f\u09ac\u0982 a<\/span> \u09ac\u09cd\u09af\u09be\u09b8\u09be\u09b0\u09cd\u09a7\u09ac\u09bf\u09b6\u09bf\u09b7\u09cd\u099f \u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3:<\/strong><\/span><\/h3>\r\n\r\n\r\n\r\n

\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf,\u00a0 \u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u0993\u09aa\u09b0 P(x,y)<\/span> \u09af\u09c7 \u0995\u09cb\u09a8\u09cb \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u098f\u09ac\u0982 C(h,k)<\/span> \u09ac\u09c3\u09a4\u09cd\u09a4\u099f\u09bf\u09b0 \u0995\u09c7\u09a8\u09cd\u09a6\u09cd\u09b0\u0964 \u098f\u0996\u09a8 PM \\perp OX, CN \\perp OX, CQ \\perp PM<\/span> \u0985\u0999\u09cd\u0995\u09a8 \u0995\u09b0\u09bf \u098f\u09ac\u0982 C,P<\/span>\u00a0\u09af\u09cb\u0997 \u0995\u09b0\u09bf\u0964<\/span><\/p>\r\n\r\n\r\n\r\n

\r\n
\"\u09ac\u09cd\u09af\u09be\u09b8\u09be\u09b0\u09cd\u09a7<\/figure>\r\n<\/div>\r\n\r\n\r\n\r\n

\u09a4\u09be\u09b9\u09b2\u09c7,<\/span> OM=x, PM=y, ON=h, CN=k<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

\\therefore CQ=NM=OM-ON=x-h<\/span> \u098f\u09ac\u0982 PQ=PM-QM=PM-CN=y-k<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

\\Delta CPQ<\/span><\/span>\u00a0\u09b9\u09a4\u09c7 \u09aa\u09be\u0987,<\/span> CQ^2+PQ^2=CP^2<\/span><\/span>\u00a0<\/span>[<\/span>\u2235\u2206CPQ <\/span>\u09b8\u09ae\u0995\u09cb\u09a3\u09c0 \u09a4\u09cd\u09b0\u09bf\u09ad\u09c1\u099c] <\/span>\u00a0<\/span><\/p>\r\n\r\n\r\n\r\n

\u0985\u09b0\u09cd\u09a5\u09be\u09ce (x-h)^2+(y-k)^2=a^2<\/span> <\/span>[<\/span>\u2235CP=a,<\/span> \u09a4\u09cd\u09b0\u09bf\u09ad\u09c1\u099c\u09c7\u09b0 \u09ac\u09cd\u09af\u09be\u09b8\u09be\u09b0\u09cd\u09a7]<\/span><\/p>\r\n\r\n\r\n\r\n

\u098f\u0987 \u09b8\u09ae\u09cd\u09aa\u09b0\u09cd\u0995\u099f\u09bf \u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u0993\u09aa\u09b0\u09b8\u09cd\u09a5 \u09af\u09c7 \u0995\u09cb\u09a8\u09cb \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09b0 \u099c\u09a8\u09cd\u09af \u09aa\u09cd\u09b0\u09af\u09cb\u099c\u09cd\u09af\u0964<\/span>\u00a0<\/span>\u09b8\u09c1\u09a4\u09b0\u09be\u0982 \u098f\u099f\u09be\u0987 \u09a8\u09bf\u09b0\u09cd\u09a3\u09c7\u09df \u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u0964<\/span><\/p>\r\n\r\n\r\n\r\n

\u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u09b8\u09be\u09a7\u09be\u09b0\u09a3 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 (General equation of a circle):<\/strong><\/span><\/h2>\r\n\r\n\r\n\r\n

(a,b)<\/span> \u0995\u09c7\u09a8\u09cd\u09a6\u09cd\u09b0 \u0993 r<\/span>\u00a0\u09ac\u09cd\u09af\u09be\u09b8\u09be\u09b0\u09cd\u09a7\u09ac\u09bf\u09b6\u09bf\u09b7\u09cd\u099f \u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3<\/strong>\u00a0\u00a0<\/span><\/span><\/h3>\r\n\r\n\r\n\r\n

(x-a)^2+(y-b)^2=r^2<\/span> \u0995\u09c7 \u09a8\u09bf\u09ae\u09cd\u09a8\u09b0\u09c2\u09aa\u09c7 \u09b2\u09c7\u0996\u09be \u09af\u09be\u09df:<\/span><\/p>\r\n\r\n\r\n\r\n

x^2-2ax+a^2+y^2-2by+b^2-r^2=0<\/span>\u00a0<\/span><\/p>\r\n\r\n\r\n\r\n

\u09ac\u09be, x^2+y^2-2ax-2by+(a^2+b^2-r^2)=0<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

\u098f\u0996\u09a8, (a,b)<\/span><\/span>\u00a0\u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1<\/span>\u00a0<\/span> \u0993 r<\/span><\/span> \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09ac\u09cd\u09af\u09be\u09b8\u09be\u09b0\u09cd\u09a7 \u09ac\u09b2\u09c7 a, b<\/span><\/span> \u0993 r<\/span><\/span> \u098f\u09ac\u0982 a^2+b^2-r^2<\/span><\/span>\u00a0\u09a7\u09cd\u09b0\u09c1\u09ac\u0995\u0964<\/span>\u00a0<\/span><\/p>\r\n\r\n\r\n\r\n

\u09a7\u09b0\u09bf, -a=g, -b=f<\/span><\/span>\u00a0\u09ac\u09be, (a,b)=(-g,-f)<\/span> <\/span>\u098f\u09ac\u0982<\/span>\u00a0<\/span> a^2+b^2-r^2=c<\/span>\u00a0<\/span><\/p>\r\n\r\n\r\n\r\n

\u09ac\u09be, (-g)^2+(-f)^2-r^2=c [\\because -a=g, -b=f]<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

\u09ac\u09be, (-g)^2+(-f)^2-c=r^2<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

\u09ac\u09be, r=\\sqrt{g^2+f^2-c}<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

\u09b8\u09c1\u09a4\u09b0\u09be\u0982, \u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u099f\u09bf \u09b9\u09b2\u09cb,<\/span> x^2+y^2+2gx+2fy+c=0<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

\u09af\u09be\u09b0 \u0995\u09c7\u09a8\u09cd\u09a6\u09cd\u09b0\u09c7\u09b0 \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995 (-g,-f)=\\Big(\\frac{x \u098f\u09b0\\; \u09b8\u09b9\u0997}{-2}, \\frac{y \u098f\u09b0\\; \u09b8\u09b9\u0997}{-2}\\Big)<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

\u098f\u09ac\u0982 \u09ac\u09cd\u09af\u09be\u09b8\u09be\u09b0\u09cd\u09a7 <\/span>=\\sqrt{g^2+f^2-c}<\/span><\/span>; \u09af\u0996\u09a8<\/span> g^2+f^2-c \\geq 0<\/span>\u00a0<\/span><\/p>\r\n\r\n\r\n\r\n

\u098f\u0995\u09c7 \u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u09b8\u09be\u09a7\u09be\u09b0\u09a3 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u09ac\u09b2\u09c7\u0964<\/span><\/p>\r\n\r\n\r\n\r\n

\u0985\u09a8\u09c1\u09b8\u09bf\u09a6\u09cd\u09a7\u09be\u09a8\u09cd\u09a4<\/b>: A \\neq 0<\/span><\/b> \u09b9\u09b2\u09c7 Ax^2+Ay^2+Dx+Ey+F=0<\/span><\/span>\u00a0\u0986\u0995\u09be\u09b0\u09c7\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u09b9\u09df<\/span>\u00a0<\/span> \u098f\u0995\u099f\u09bf \u09ac\u09c3\u09a4\u09cd\u09a4 \u09a8\u09bf\u09b0\u09cd\u09a6\u09c7\u09b6 \u0995\u09b0\u09c7 \u0985\u09a5\u09ac\u09be \u098f\u09b0 \u0995\u09cb\u09a8\u09cb \u09b2\u09c7\u0996\u099a\u09bf\u09a4\u09cd\u09b0 \u09a8\u09c7\u0987\u0964 \u098f\u0995\u09c7\u0993 \u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u09b8\u09be\u09a7\u09be\u09b0\u09a3 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u09ac\u09b2\u09be \u09b9\u09df\u0964<\/span><\/p>\r\n\r\n\r\n\r\n

\u09ac\u09cd\u09af\u09be\u0996\u09cd\u09af\u09be:<\/b> Ax^2+Ay^2+Dx+Ey+F=0<\/span> \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u09b9\u09a4\u09c7 \u09aa\u09be\u0987,<\/span>\u00a0<\/span><\/p>\r\n\r\n\r\n\r\n

\u09ac\u09be, A(x^2+y^2)+Dx+Ey+F=0<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

\u09ac\u09be, x^2+y^2+\\frac{D}{A}x+\\frac{E}{A}y+\\frac{F}{A}=0<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

\u09ac\u09be, x^2+y^2+2gx+2fy+c=0<\/span><\/span> \u0986\u0995\u09be\u09b0\u09c7\u09b0 \u09ac\u09b2\u09c7 \u098f\u0995\u099f\u09bf \u09ac\u09c3\u09a4\u09cd\u09a4 \u09a8\u09bf\u09b0\u09cd\u09a6\u09c7\u09b6 \u0995\u09b0\u09c7 \u09af\u0996\u09a8 g^2+f^2-c \\geq 0<\/span><\/span>\u00a0\u09b9\u09df\u0964<\/span><\/p>\r\n\r\n\r\n\r\n

\u09af\u09a6\u09bf g^2+f^2-c < 0<\/span><\/span> \u09b9\u09df, \u09a4\u09ac\u09c7 x^2+y^2+2gx+2fy+c=0<\/span><\/span>\u00a0\u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u099f\u09bf \u0995\u09cb\u09a8\u09cb \u09ac\u09be\u09b8\u09cd\u09a4\u09ac \u09b8\u099e\u09cd\u099a\u09be\u09b0\u09aa\u09a5 \u09a8\u09bf\u09b0\u09cd\u09a6\u09c7\u09b6 \u0995\u09b0\u09c7 \u09a8\u09be \u098f\u09ac\u0982 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u099f\u09bf\u09b0 \u0995\u09cb\u09a8\u09cb \u09b2\u09c7\u0996\u099a\u09bf\u09a4\u09cd\u09b0 \u0985\u0999\u09cd\u0995\u09a8 \u0995\u09b0\u09be \u09af\u09be\u09df \u09a8\u09be\u0964<\/span><\/p>\r\n\r\n\r\n\r\n

\u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u09b8\u09be\u09a7\u09be\u09b0\u09a3 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3-\u098f\u09b0 \u09ac\u09c8\u09b6\u09bf\u09b7\u09cd\u099f\u09cd\u09af\u0997\u09c1\u09b2\u09bf<\/span>\u00a0\u09b9\u09b2\u09cb:<\/span><\/p>\r\n\r\n\r\n\r\n

    \r\n
  1. \u00a0\u098f\u099f\u09bf \u09a6\u09c1\u0987 \u099a\u09be\u09b2\u0995\u09c7\u09b0 \u09a6\u09cd\u09ac\u09bf\u0998\u09be\u09a4 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 (x^2, y^2<\/span><\/span> \u09b8\u09ae\u09cd\u09ac\u09b2\u09bf\u09a4)<\/span><\/li>\r\n
  2. xy<\/span><\/span>\u00a0\u09b8\u09ae\u09cd\u09ac\u09b2\u09bf\u09a4 \u09aa\u09a6 \u09a8\u09be\u0987, \u0985\u09b0\u09cd\u09a5\u09be\u09ce<\/span> xy<\/span><\/span>\u00a0\u098f\u09b0 \u09b8\u09b9\u0997<\/span>=0<\/span><\/li>\r\n
  3. \u00a0x^2<\/span><\/span> \u098f\u09b0 \u09b8\u09b9\u0997 <\/span>= y^2<\/span><\/span> \u098f\u09b0 \u09b8\u09b9\u0997<\/span><\/li>\r\n<\/ol>\r\n\r\n\r\n\r\n

    \u09ac\u09bf:\u09a6\u09cd\u09b0:<\/b><\/p>\r\n\r\n\r\n\r\n

      \r\n
    1. g^2+f^2-c=0<\/span> \u09ac\u09be,<\/span>\u00a0<\/span>\u09ac\u09cd\u09af\u09be\u09b8\u09be\u09b0\u09cd\u09a7 \u09b6\u09c2\u09a8\u09cd\u09af \u09b9\u09b2\u09c7 \u09ac\u09c3\u09a4\u09cd\u09a4\u099f\u09bf \u098f\u0995\u099f\u09bf \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09b8\u09b0\u09cd\u09ac\u09b8\u09cd\u09ac \u09b9\u09df\u0964 \u098f\u09b0\u09c2\u09aa \u09ac\u09c3\u09a4\u09cd\u09a4\u0995\u09c7 \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09ac\u09c3\u09a4\u09cd\u09a4 \u09ac\u09b2\u09c7\u0964<\/span><\/li>\r\n
    2. x^2+y^2+2gx+2fy+c=0<\/span> \u09ac\u09c3\u09a4\u09cd\u09a4\u099f\u09bf \u09ae\u09c2\u09b2\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 <\/span>(0,\u00a00)<\/span> \u09a6\u09bf\u09df\u09c7 \u09af\u09be\u09ac\u09c7 \u09af\u09a6\u09bf \u098f\u09ac\u0982 \u0995\u09c7\u09ac\u09b2 \u09af\u09a6\u09bf c=0<\/span><\/span> \u09b9\u09df\u0964 \u0985\u09b0\u09cd\u09a5\u09be\u09ce \u09ae\u09c2\u09b2\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u0997\u09be\u09ae\u09c0 \u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3, x^2+y^2+2gx+2fy=0<\/span><\/span><\/li>\r\n
    3. c\\leq 0<\/span><\/span> \u09b9\u09b2\u09c7, x^2+y^2+2gx+2fy+c=0<\/span><\/span> \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u099f\u09bf \u09b8\u09b0\u09cd\u09ac\u09a6\u09be\u0987 \u098f\u0995\u099f\u09bf \u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u09b8\u09c2\u099a\u09bf\u09a4 \u0995\u09b0\u09c7\u0964 \u0995\u09bf\u09a8\u09cd\u09a4\u09c1 c>0<\/span><\/span>\u00a0\u09b9\u09b2\u09c7,<\/span>\u00a0<\/span>\u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u09a8\u09be\u0993 \u09b9\u09a4\u09c7 \u09aa\u09be\u09b0\u09c7\u0964<\/span><\/li>\r\n
    4. x^2+y^2+2gx+2fy+c=0<\/span> \u09ac\u09c3\u09a4\u09cd\u09a4\u099f\u09bf\u09b0 \u0995\u09c7\u09a8\u09cd\u09a6\u09cd\u09b0 (-g, -f)<\/span><\/span>\u00a0x<\/span>-\u0985\u0995\u09cd\u09b7\u09c7\u09b0<\/span>\u00a0<\/span> \u0989\u09aa\u09b0 \u0985\u09ac\u09b8\u09cd\u09a5\u09bf\u09a4 \u09b9\u09b2\u09c7,<\/span> f=0<\/span><\/span>\u00a0\u09ac\u09be, y<\/span><\/span> \u098f\u09b0 \u09b8\u09b9\u0997 =0<\/span><\/span>\u00a0\u098f\u09ac\u0982<\/span> y<\/span><\/span>\u00a0\u0985\u0995\u09cd\u09b7\u09c7\u09b0 \u0989\u09aa\u09b0 \u0985\u09ac\u09b8\u09cd\u09a5\u09bf\u09a4 \u09b9\u09b2\u09c7,<\/span> g=0<\/span><\/span> \u09ac\u09be, x<\/span><\/span>-\u098f\u09b0 \u09b8\u09b9\u0997 =0<\/span><\/span>\u0964<\/span><\/li>\r\n<\/ol>\r\n

       <\/p>\r\n\r\n\r\n\r\n

      \u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09c7\u09b0 \u09ac\u09c8\u09b6\u09bf\u09b7\u09cd\u099f\u09cd\u09af\u09be\u09ac\u09b2\u09bf <\/b>(Features of the circle equation)<\/b>:<\/b><\/span><\/h2>\r\n\r\n\r\n\r\n
        \r\n
      1. \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u099f\u09bf x<\/span><\/span> \u0993 y<\/span><\/span> \u098f\u09b0 \u098f\u0995\u099f\u09bf \u09a6\u09cd\u09ac\u09bf\u0998\u09be\u09a4 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u0964<\/span>\u00a0<\/span><\/li>\r\n
      2. \u00a0\u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09c7 x^2<\/span><\/span>\u00a0<\/span>\u0993 y^2<\/span><\/span> \u098f\u09b0 \u09b8\u09b9\u0997 \u09b8\u09ae\u09be\u09a8\u0964<\/span><\/li>\r\n
      3. xy<\/span><\/span>\u00a0\u09b8\u09ae\u09cd\u09ac\u09b2\u09bf\u09a4 \u09aa\u09a6 \u0985\u09a8\u09c1\u09aa\u09b8\u09cd\u09a5\u09bf\u09a4\u0964<\/span><\/li>\r\n<\/ol>\r\n

         <\/p>\r\n\r\n\r\n\r\n

        \u09a6\u09cd\u09b0\u09b7\u09cd\u099f\u09ac\u09cd\u09af:<\/b><\/span><\/h3>\r\n\r\n\r\n\r\n
          \r\n
        1. g=0<\/span><\/span> \u09b9\u09b2\u09c7, \u0995\u09c7\u09a8\u09cd\u09a6\u09cd\u09b0\u09c7\u09b0 \u09ad\u09c1\u099c \u09b6\u09c2\u09a8\u09cd\u09af \u09b9\u09ac\u09c7, \u0985\u09b0\u09cd\u09a5\u09be\u09ce \u0995\u09c7\u09a8\u09cd\u09a6\u09cd\u09b0 y<\/span><\/span>-\u0985\u0995\u09cd\u09b7\u09c7\u09b0 \u0993\u09aa\u09b0 \u0985\u09ac\u09b8\u09cd\u09a5\u09bf\u09a4 \u09b9\u09ac\u09c7\u0964<\/span><\/li>\r\n
        2. f=0<\/span><\/span>\u00a0\u09b9\u09b2\u09c7,<\/span>\u00a0<\/span> \u0995\u09c7\u09a8\u09cd\u09a6\u09cd\u09b0\u09c7\u09b0 \u0995\u09cb\u099f\u09bf \u09b6\u09c2\u09a8\u09cd\u09af \u09b9\u09ac\u09c7, \u0985\u09b0\u09cd\u09a5\u09be\u09ce \u0995\u09c7\u09a8\u09cd\u09a6\u09cd\u09b0 x<\/span><\/span>-\u0985\u0995\u09cd\u09b7\u09c7\u09b0<\/span>\u00a0<\/span> \u00a0<\/span>\u0993\u09aa\u09b0 \u0985\u09ac\u09b8\u09cd\u09a5\u09bf\u09a4 \u09b9\u09ac\u09c7\u0964<\/span>\u00a0<\/span><\/li>\r\n
        3. c=0<\/span><\/span> \u09b9\u09b2\u09c7, \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u099f\u09bf \u09ae\u09c2\u09b2\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 (0,0)<\/span><\/span>\u00a0\u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09b8\u09bf\u09a6\u09cd\u09a7 \u09b9\u09ac\u09c7, \u0985\u09b0\u09cd\u09a5\u09be\u09ce \u09ac\u09c3\u09a4\u09cd\u09a4\u099f\u09bf \u09ae\u09c2\u09b2\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09a6\u09bf\u09df\u09c7 \u09af\u09be\u09ac\u09c7\u0964<\/span><\/li>\r\n
        4. \u00a0\u09af\u09a6\u09bf \u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u09ac\u09cd\u09af\u09be\u09b8\u09be\u09b0\u09cd\u09a7<\/span>\u00a0<\/span>=<\/span> \u0995\u09c7\u09a8\u09cd\u09a6\u09cd\u09b0\u09c7\u09b0 \u0995\u09cb\u099f\u09bf\u09b0 \u09aa\u09b0\u09ae\u09ae\u09be\u09a8 \u0985\u09b0\u09cd\u09a5\u09be\u09ce,<\/span> \\sqrt{g^2+f^2-c}=|f|<\/span><\/span> \u09b9\u09df,<\/span> \u09a4\u09ac\u09c7 \u09ac\u09c3\u09a4\u09cd\u09a4\u099f\u09bf x<\/span>-\u0985\u0995\u09cd\u09b7\u0995\u09c7 \u09b8\u09cd\u09aa\u09b0\u09cd\u09b6 \u0995\u09b0\u09ac\u09c7\u0964 \u09b8\u09c7\u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7 g^2=c<\/span><\/li>\r\n
        5. \u09af\u09a6\u09bf \u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u09ac\u09cd\u09af\u09be\u09b8\u09be\u09b0\u09cd\u09a7 <\/span>=<\/span> \u0995\u09c7\u09a8\u09cd\u09a6\u09cd\u09b0\u09c7\u09b0 \u09ad\u09c1\u099c\u09c7\u09b0 \u09aa\u09b0\u09ae\u09ae\u09be\u09a8 \u0985\u09b0\u09cd\u09a5\u09be\u09ce, \\sqrt{g^2+f^2-c}=|g|<\/span><\/span> \u09b9\u09df, <\/span>\u09a4\u09ac\u09c7 \u09ac\u09c3\u09a4\u09cd\u09a4\u099f\u09bf y<\/span>-\u0985\u0995\u09cd\u09b7\u0995\u09c7 \u09b8\u09cd\u09aa\u09b0\u09cd\u09b6 \u0995\u09b0\u09ac\u09c7\u0964 \u09b8\u09c7\u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7 f^2=c<\/span><\/li>\r\n
        6. \u09ac\u09c3\u09a4\u09cd\u09a4\u099f\u09bf \u0989\u09ad\u09df\u09aa\u0995\u09cd\u09b7\u0995\u09c7 \u09b8\u09cd\u09aa\u09b0\u09cd\u09b6 \u0995\u09b0\u09b2\u09c7 g^2=f^2=c<\/span>\u0964<\/li>\r\n<\/ol>\r\n

           <\/p>\r\n\r\n\r\n\r\n

          \u09a6\u09c1\u0987\u099f\u09bf \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u0995\u09c7 \u09ac\u09cd\u09af\u09be\u09b8\u09c7\u09b0 \u09aa\u09cd\u09b0\u09be\u09a8\u09cd\u09a4\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09a7\u09b0\u09c7 \u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 <\/b>(Equation of the circle along the vertex of the diameter at two fixed points)<\/b>:<\/b><\/span><\/h2>\r\n\r\n\r\n\r\n

          \u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, \u09b8\u09ae\u09a4\u09b2\u09c7 \u09a6\u09c1\u0987\u099f\u09bf \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 A(x_1,y_1)<\/span> \u0993 B(x_2,y_2)<\/span> \u098f\u09ac\u0982 AB<\/span>\u0995\u09c7 \u09ac\u09cd\u09af\u09be\u09b8 \u09a7\u09b0\u09c7 \u0985\u0999\u09cd\u0995\u09bf\u09a4 \u09ac\u09c3\u09a4\u09cd\u09a4\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\u0964 \u09af\u09c7\u09b9\u09c7\u09a4\u09c1 \u0985\u09b0\u09cd\u09a7\u09ac\u09c3\u09a4\u09cd\u09a4\u09b8\u09cd\u09a5 \u0995\u09cb\u09a3 \u098f\u0995 \u09b8\u09ae\u0995\u09cb\u09a3, \u09a4\u09be\u0987 P<\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u099f\u09bf \u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u0989\u09aa\u09b0 \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09a8 \u0995\u09b0\u09ac\u09c7 \u09af\u09a6\u09bf \u098f\u09ac\u0982 \u0995\u09c7\u09ac\u09b2 \u09af\u09a6\u09bf \\angle APB =1<\/span>\u00a0\u09b8\u09ae\u0995\u09cb\u09a3 \u09b9\u09df\u0964\u00a0<\/span><\/p>\r\n\r\n\r\n\r\n

          \r\n
          \"\u09ac\u09cd\u09af\u09be\u09b8\u09be\u09b0\u09cd\u09a7<\/figure>\r\n<\/div>\r\n\r\n\r\n\r\n

          \u09b8\u09c1\u09a4\u09b0\u09be\u0982<\/span> AP \\perp BP<\/span><\/span><\/p>\r\n\r\n\r\nx^2+y^2+2gx+2fy+c=0\\\\ \\sqrt{g^2+f^2-c}=|g|\\\\ \\angle APB =1\\\\ \\Rightarrow AP-\u098f\u09b0\\; \u09a2\u09be\u09b2 \\times BP-\u098f\u09b0\\; \u09a2\u09be\u09b2 =-1\\\\ \\Rightarrow \\frac{y-y_1}{x-x_1}\\times \\frac{y-y2}{x-x_2}=-1\\\\ \\Rightarrow (y-y_1)(y-y_2)=-(x-x_1)(x-x_2)\\\\ \\therefore (x-x_1)(x-x_2)+(y-y_1)(y-y_2)=0\u00a0<\/span>\n\r\n\r\n\r\n

          \\therefore (x_1,y_1)<\/span> \u0993 (x_2,y_2)<\/span><\/span>\u00a0\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a6\u09cd\u09ac\u09df\u0995\u09c7<\/span>\u00a0<\/span> \u09ac\u09cd\u09af\u09be\u09b8\u09c7\u09b0 \u09aa\u09cd\u09b0\u09be\u09a8\u09cd\u09a4\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09a7\u09b0\u09c7 \u0985\u0999\u09cd\u0995\u09bf\u09a4 \u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3: (x-x_1)(x-x_2)+(y-y_1)(y-y_2)=0<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

          \u098f\u0995\u099f\u09bf \u09ac\u09c3\u09a4\u09cd\u09a4 \u0993 \u098f\u0995\u099f\u09bf \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u099b\u09c7\u09a6\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09a6\u09bf\u09df\u09c7 \u09af\u09be\u09df \u098f\u09b0\u09c2\u09aa \u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u09a8\u09bf\u09b0\u09cd\u09a3\u09df \u0995\u09b0\u09a4\u09c7 \u09b9\u09ac\u09c7\u0964\u00a0<\/b><\/span><\/h3>\r\n\r\n\r\n\r\n
          \r\n
          \"\u09ac\u09cd\u09af\u09be\u09b8\u09be\u09b0\u09cd\u09a7<\/figure>\r\n<\/div>\r\n\r\n\r\n\r\n

          \u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, \u09ac\u09c3\u09a4\u09cd\u09a4\u099f\u09bf\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 x^2+y^2+2gx+2fy+c=0......(i)<\/span> <\/span>\u098f\u09ac\u0982 \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u099f\u09bf\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 lx+my+n=0......(ii)<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

          k<\/span><\/span> \u098f\u0995\u099f\u09bf \u0987\u099a\u09cd\u099b\u09be\u09ae\u09c2\u09b2\u0995 \u09a7\u09cd\u09b0\u09c1\u09ac\u0995 (k\\neq 0)<\/span><\/span> \u09b9\u09b2\u09c7, (i)+k(ii)<\/span><\/span>\u00a0\u09b9\u09a4\u09c7 \u09aa\u09be\u0987,<\/span>\u00a0<\/span><\/p>\r\n\r\n\r\n\r\n

          x^2+y^2+2gx+2fy+c+k(lx+my+n)=0......(iii)<\/span>\u00a0<\/span><\/p>\r\n\r\n\r\n\r\n

          \\Rightarrow x^2+y^2+(2g+kl)x+(2f+km)y+c+kn=0<\/span>;<\/span> \u09af\u09be \u098f\u0995\u099f\u09bf \u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u09b8\u0995\u09b2 \u09b6\u09b0\u09cd\u09a4 \u09b8\u09bf\u09a6\u09cd\u09a7 \u0995\u09b0\u09c7\u0964\u00a0<\/span><\/p>\r\n\r\n\r\n\r\n

          \u098f\u0995\u099f\u09bf \u09ac\u09c3\u09a4\u09cd\u09a4 \u0993 \u098f\u0995\u099f\u09bf \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u099b\u09c7\u09a6\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u0997\u09be\u09ae\u09c0 \u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3, \u09ac\u09c3\u09a4\u09cd\u09a4 <\/span>+k<\/span><\/span>(\u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be) <\/span>=0<\/span><\/p>\r\n\r\n\r\n\r\n

          \u0985\u09a8\u09c1\u09b0\u09c2\u09aa\u09ad\u09be\u09ac\u09c7, \u09a6\u09c1\u0987\u099f\u09bf \u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u099b\u09c7\u09a6\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09a6\u09bf\u09df\u09c7 \u09af\u09be\u09df \u098f\u09b0\u09c2\u09aa \u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3,<\/strong><\/p>\r\n\r\n\r\n\r\n

          \u09aa\u09cd\u09b0\u09a5\u09ae \u09ac\u09c3\u09a4\u09cd\u09a4 +k<\/span>(\u09a6\u09cd\u09ac\u09bf\u09a4\u09c0\u09df \u09ac\u09c3\u09a4\u09cd\u09a4) =0<\/span>;<\/span> k<\/span> \u098f\u0995\u099f\u09bf \u0987\u099a\u09cd\u099b\u09be\u09ae\u09c2\u09b2\u0995 \u09a7\u09cd\u09b0\u09c1\u09ac\u0995 (k\\neq 0)<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

          \r\n
          \"\u09a6\u09c1\u0987\u099f\u09bf<\/figure>\r\n<\/div>\r\n\r\n\r\n\r\n

          \\therefore f(x,y)=0<\/span> \u09ac\u09c3\u09a4\u09cd\u09a4 \u0993 g(x,y)=0<\/span><\/span> \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 (\u0985\u09a5\u09ac\u09be, f(x,y)=0<\/span><\/span>\u00a0\u0993 g(x,y)=0<\/span><\/span>\u00a0\u09ac\u09c3\u09a4\u09cd\u09a4\u09a6\u09cd\u09ac\u09df\u09c7\u09b0) \u099b\u09c7\u09a6\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u098f\u09ac\u0982<\/span> (\\alpha, \\beta)<\/span><\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u0997\u09be\u09ae\u09c0 \u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \\frac{f(x,y)}{f(\\alpha, \\beta)}=\\frac{g(x,y)}{a(\\alpha, \\beta)}, f(\\alpha, \\beta)\\neq 0, g(\\alpha, \\beta)\\neq 0<\/span><\/span><\/p>\r\n

           <\/p>\r\n

          \r\n

           <\/p>\r\n

          \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>\r\n