{"id":1011,"date":"2024-01-30T12:18:09","date_gmt":"2024-01-30T06:18:09","guid":{"rendered":"https:\/\/stage-wp.10minuteschool.com\/?p=1011"},"modified":"2024-11-03T20:48:39","modified_gmt":"2024-11-03T14:48:39","slug":"intersection-of-two-circle-intersecting-points","status":"publish","type":"post","link":"https:\/\/10minuteschool.com\/content\/intersection-of-two-circle-intersecting-points\/","title":{"rendered":"\u09a6\u09c1\u0987\u099f\u09bf \u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\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 (Intersection of two circle intersecting points)"},"content":{"rendered":"\r\n

\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3: <\/b><\/span>\u099b\u09c7\u09a6\u0997\u09be\u09ae\u09c0 <\/b><\/span><\/b>\u09a6\u09c1\u0987\u099f\u09bf \u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 <\/b>(Intersection of two circle intersecting points)<\/b><\/span><\/h2>\r\n\r\n\r\n\r\n

\u0986\u0997\u09c7\u09b0 \u0985\u09a8\u09c1\u099a\u09cd\u099b\u09c7\u09a6\u09c7\u09b0 \u09ae\u09a4\u09cb \u09af\u09c1\u0995\u09cd\u09a4\u09bf \u09aa\u09cd\u09b0\u09a6\u09b0\u09cd\u09b6\u09a8 \u0995\u09b0\u09c7 \u09a6\u09c7\u0996\u09be\u09a8\u09cb \u09af\u09be\u09df \u09af\u09c7, k<\/span><\/span> \u09af\u09c7<\/span>\u00a0<\/span> \u0995\u09cb\u09a8\u09cb \u098f\u0995\u099f\u09bf \u09a7\u09cd\u09b0\u09c1\u09ac\u0995 \u09b9\u09b2\u09c7, A_1x^2+A_1y^2+D_1x+E_1y+F_1=0<\/span><\/span> \u0993 A_2x^2+A_2y^2+D_2x+E_2y+F2=0<\/span><\/span>\u00a0\u09ac\u09c3\u09a4\u09cd\u09a4 \u09a6\u09c1\u0987\u099f\u09bf\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,<\/span> A_1x^22+A_1y^2+D_1x+E_1y+F_1+k(A_2x^2+A_2y^2+D_2x+E_2y+F_2)=0\u00a0<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

\u0985\u09b0\u09cd\u09a5\u09be\u09ce,<\/span>\u00a0<\/span>\u09aa\u09cd\u09b0\u09a5\u09ae \u09ac\u09c3\u09a4\u09cd\u09a4 +k<\/span><\/span>(\u09a6\u09cd\u09ac\u09bf\u09a4\u09c0\u09df \u09ac\u09c3\u09a4\u09cd\u09a4) =0<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

\u098f, \u0986\u09b0, \u0996\u09b2\u09bf\u09ab\u09be (\u0986\u099c\u09bf\u099c\u09c1\u09b0 \u09b0\u09b9\u09ae\u09be\u09a8 \u0996\u09b2\u09bf\u09ab\u09be) \u098f\u09b0 \u09a8\u09bf\u09df\u09ae\u09be\u09a8\u09c1\u09b8\u09be\u09b0\u09c7 \u09af\u09c7 \u0995\u09cb\u09a8\u09cb\u00a0 \u09a6\u09c1\u0987\u099f\u09bf \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 <\/b>(x_1, y_1)<\/span> \u0993 (x_2,y_2)<\/span>\u00a0\u09a6\u09bf\u09df\u09c7 \u0985\u09a4\u09bf\u0995\u09cd\u09b0\u09ae \u0995\u09b0\u09c7 \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><\/p>\r\n\r\n\r\n\r\n

\r\n
\"\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1<\/figure>\r\n<\/div>\r\n\r\n\r\n\r\n

(x_1,\u00a0y_1) \u0993 (x_2,y_2)<\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a6\u09cd\u09ac\u09df\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,<\/span> (x-x_1)(x-x_2)+(y-y_1)(y-y_2)=0......(i)\u00a0<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

\u0986\u09ac\u09be\u09b0, (x_1,\u00a0y_1) \u0993 (x_2,y_2)<\/span><\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u0997\u09be\u09ae\u09c0 \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3, \\Rightarrow (x-x_1)(y_1-y_2)-(y-y_1)(x_1-x_2)=0......(ii)\u00a0<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

\u2234(i)<\/span> \u09ac\u09c3\u09a4\u09cd\u09a4<\/a><\/span> \u098f\u09ac\u0982 <\/span>(ii)<\/span> \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<\/a><\/span> \u09b8\u09ae\u09c0\u0995\u09b0\u09a3,<\/span><\/p>\r\n\r\n\r\n(x-x_1)(x-x_2)+(y-y_1)(y-y_2)+k{(x-x_1)(y_1-y_2)-(y-y_1)(x_1-x_2)}=0<\/span>\n\r\n\r\n\r\n

x^2+y^2+2gx+2fy+c=0......(1),\u00a0(g^2>c,f^2>c)<\/span> \u098f\u09ac\u0982 (x-h)^2+(y-k)^2=r^2.....(2),(r^2>h^2,r^2>k^2)<\/span>\u00a0\u09ac\u09c3\u09a4\u09cd\u09a4 \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u0985\u0995\u09cd\u09b7\u09a6\u09cd\u09ac\u09df\u09c7\u09b0 \u0996\u09a8\u09cd\u09a1\u09bf\u09a4 \u0985\u0982\u09b6\u09c7\u09b0 \u09aa\u09b0\u09bf\u09ae\u09be\u09a3:<\/strong><\/span><\/h2>\r\n\r\n\r\n\r\n

(1) \u09ac\u09c3\u09a4\u09cd\u09a4\u099f\u09bf x<\/span>-\u0985\u0995\u09cd\u09b7\u0995\u09c7 \u0985\u09b0\u09cd\u09a5\u09be\u09ce, y=0<\/span>\u00a0\u09b0\u09c7\u0996\u09be\u0995\u09c7 \u09af\u09c7 \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 \u099b\u09c7\u09a6 \u0995\u09b0\u09c7 \u09a4\u09be\u09b0 \u09ad\u09c1\u099c<\/span> x^2+2gx+c=0......(3)<\/span> \u09a8\u0982 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u09b9\u09a4\u09c7 \u09aa\u09be\u0993\u09df\u09be \u09af\u09be\u09df\u0964 \u09ac\u09c3\u09a4\u09cd\u09a4\u099f\u09bf x<\/span>-\u0985\u0995\u09cd\u09b7\u0995\u09c7 A(x_1,\u00a00) \u0993 B(x_2,0)<\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 \u099b\u09c7\u09a6 \u0995\u09b0\u09c7 \u0985\u09b0\u09cd\u09a5\u09be\u09ce, (2) \u098f\u09b0 \u09ae\u09c2\u09b2 \u09a6\u09c1\u0987\u099f\u09bf x_1<\/span> \u0993 x_2<\/span> \u09b9\u09b2\u09c7 x_1+x_2=-2g<\/span>\u00a0\u098f\u09ac\u0982 x_1x_2=c<\/span>\u00a0<\/span><\/p>\r\n\r\n\r\n\r\n

\r\n
\"\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1<\/figure>\r\n<\/div>\r\n\r\n\r\n\r\n

\u2234<\/span> \u09ac\u09c3\u09a4\u09cd\u09a4\u099f\u09bf \u09a6\u09cd\u09ac\u09be\u09b0\u09be x<\/span><\/span>-\u0985\u0995\u09cd\u09b7\u09c7\u09b0 \u099b\u09c7\u09a6\u09be\u0982\u09b6\u09c7\u09b0 \u09aa\u09b0\u09bf\u09ae\u09be\u09a3 =AB=OB-OA<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

\\Rightarrow x_2-x_1=\\sqrt{(x_1+x_2)^2-4x_1x_2}=\\sqrt{4g^2-4c}<\/span>\u00a0<\/span><\/p>\r\n\r\n\r\n\r\n

\\Rightarrow 2\\sqrt{g^2-c}=2\\sqrt{h^2-(h^2+k^2-r^2)}=2\\sqrt{r^2-k^2}<\/span>\u00a0<\/span><\/p>\r\n\r\n\r\n\r\n

\u0985\u09a8\u09c1\u09b0\u09c2\u09aa\u09ad\u09be\u09ac\u09c7, y<\/span><\/span>-\u0985\u0995\u09cd\u09b7\u09c7\u09b0 \u099b\u09c7\u09a6\u09be\u0982\u09b6\u09c7\u09b0 \u09aa\u09b0\u09bf\u09ae\u09be\u09a3 =2\\sqrt{f^2-c}=2\\sqrt{r^2-h^2}<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

\u09ac\u09bf:\u09a6\u09cd\u09b0:\u00a0<\/b><\/p>\r\n\r\n\r\n(x-h)^2+(y-k)^2=r^2......(1),\u00a0\u00a0x^2+y^2+2gx+2fy+c=0......(2)<\/span>\n\r\n\r\n\r\n

\r\n
\"\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1<\/figure>\r\n<\/div>\r\n\r\n\r\n\r\n
    \r\n
  1. (1) \u0993 (2) \u09ac\u09c3\u09a4\u09cd\u09a4\u09a6\u09cd\u09ac\u09df \u09ae\u09c2\u09b2\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09a6\u09bf\u09df\u09c7 \u0985\u09a4\u09bf\u0995\u09cd\u09b0\u09ae \u0995\u09b0\u09b2\u09c7, r^2=h^2+k^2,\u00a0c=0<\/span>\u00a0(\u099a\u09bf\u09a4\u09cd\u09b0- \u09e7)\u0964\u00a0\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\u0995\u09c7 \u09b2\u09bf\u0996\u09be \u09af\u09be\u09df, x^2+y^2+2gx+2fy=0,\u00a0\u00a0x^2+y^2-2hx-2ky=0<\/span><\/span><\/li>\r\n
  2. (1) \u09ac\u09c3\u09a4\u09cd\u09a4\u099f\u09bf X<\/span>-\u0985\u0995\u09cd\u09b7\u0995\u09c7 \u09b8\u09cd\u09aa\u09b0\u09cd\u09b6 \u0995\u09b0\u09b2\u09c7,\u00a0\u09ac\u09cd\u09af\u09be\u09b8\u09be\u09b0\u09cd\u09a7, r<\/span>=|\u0995\u09c7\u09a8\u09cd\u09a6\u09cd\u09b0\u09c7\u09b0 \u0995\u09cb\u099f\u09bf|, \u09ab\u09b2\u09c7 \u09ac\u09c3\u09a4\u09cd\u09a4\u099f\u09bf\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 (x-h)^2+(y-k)^2=k^2<\/span>;<\/span> (2)\u00a0\u09ac\u09c3\u09a4\u09cd\u09a4\u099f\u09bf \u0985\u0995\u09cd\u09b7\u0995\u09c7 \u09b8\u09cd\u09aa\u09b0\u09cd\u09b6 \u0995\u09b0\u09b2\u09c7, 2\\sqrt{g^2-c}=0\\Rightarrow g^2=c<\/span>\u0964 <\/span>\u09b8\u09cd\u09aa\u09b0\u09cd\u09b6\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09b0 \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995 (h,0)<\/span> \u09ac\u09be, (-g,0)<\/span>\u00a0(\u099a\u09bf\u09a4\u09cd\u09b0-\u09e8)<\/li>\r\n
  3. (1) \u09ac\u09c3\u09a4\u09cd\u09a4\u099f\u09bf y<\/span>-\u0985\u0995\u09cd\u09b7\u0995\u09c7 \u09b8\u09cd\u09aa\u09b0\u09cd\u09b6 \u0995\u09b0\u09b2\u09c7,\u00a0\u09ac\u09cd\u09af\u09be\u09b8\u09be\u09b0\u09cd\u09a7, r<\/span>=|\u0995\u09c7\u09a8\u09cd\u09a6\u09cd\u09b0\u09c7\u09b0 \u09ad\u09c1\u099c|, \u09ab\u09b2\u09c7 \u09ac\u09c3\u09a4\u09cd\u09a4\u099f\u09bf\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 (x-h)^2+(y-h)^2=h^2<\/span>;<\/span> (2)\u00a0\u09ac\u09c3\u09a4\u09cd\u09a4\u099f\u09bf \u0985\u0995\u09cd\u09b7\u0995\u09c7 \u09b8\u09cd\u09aa\u09b0\u09cd\u09b6 \u0995\u09b0\u09b2\u09c7, 2\\sqrt{f^2-c}=0\\Rightarrow f^2=c<\/span>\u0964 \u09b8\u09cd\u09aa\u09b0\u09cd\u09b6\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09b0 \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995 (0,k)<\/span> \u09ac\u09be, (0,-f)<\/span>\u00a0(\u099a\u09bf\u09a4\u09cd\u09b0-\u09e9)\u00a0<\/span><\/li>\r\n
  4. (1) \u0993 (2) \u09ac\u09c3\u09a4\u09cd\u09a4\u09a6\u09cd\u09ac\u09df \u0989\u09ad\u09df \u0985\u0995\u09cd\u09b7\u0995\u09c7 \u09b8\u09cd\u09aa\u09b0\u09cd\u09b6 \u0995\u09b0\u09b2\u09c7, r=|h|=|k|<\/span> \u098f\u09ac\u0982 g^2=f^2=c<\/span>\u00a0(\u099a\u09bf\u09a4\u09cd\u09b0-\u09ea)\u0964 \u098f\u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7 \u0995\u09c7\u09a8\u09cd\u09a6\u09cd\u09b0 \u09b9\u09ac\u09c7 (\\pm h, \\pm h)<\/span><\/span><\/li>\r\n
  5. (1) \u0993 (2) \u09ac\u09c3\u09a4\u09cd\u09a4\u09a6\u09cd\u09ac\u09df\u09c7\u09b0 \u0995\u09c7\u09a8\u09cd\u09a6\u09cd\u09b0 x<\/span>-\u0985\u0995\u09cd\u09b7\u09c7\u09b0 \u0989\u09aa\u09b0 \u0985\u09ac\u09b8\u09cd\u09a5\u09bf\u09a4 \u09b9\u09b2\u09c7, k=0,f=0<\/span>, \u09ab\u09b2\u09c7 \u09ac\u09c3\u09a4\u09cd\u09a4\u09a6\u09cd\u09ac\u09df\u09c7\u09b0\u00a0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u09b9\u09ac\u09c7 (x-h)^2+y^2=r^2, x^2+y^2+2gx+c=0<\/span>\u00a0(\u099a\u09bf\u09a4\u09cd\u09b0-\u09eb)<\/span><\/li>\r\n
  6. (1) \u0993 (2) \u09ac\u09c3\u09a4\u09cd\u09a4\u09a6\u09cd\u09ac\u09df\u09c7\u09b0 \u0995\u09c7\u09a8\u09cd\u09a6\u09cd\u09b0 y<\/span>-\u0985\u0995\u09cd\u09b7\u09c7\u09b0 \u0989\u09aa\u09b0 \u0985\u09ac\u09b8\u09cd\u09a5\u09bf\u09a4 \u09b9\u09b2\u09c7, h=0, g=0<\/span>, \u09ab\u09b2\u09c7 \u09ac\u09c3\u09a4\u09cd\u09a4\u09a6\u09cd\u09ac\u09df\u09c7\u09b0\u00a0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u09b9\u09ac\u09c7 x^2+(y-k)^2=r^2,\u00a0x^2+y^2+2fy+c=0<\/span>\u00a0(\u099a\u09bf\u09a4\u09cd\u09b0-\u09ec)<\/span><\/li>\r\n
  7. (1) \u0993 (2) \u09ac\u09c3\u09a4\u09cd\u09a4\u09a6\u09cd\u09ac\u09df y<\/span>-\u0985\u0995\u09cd\u09b7\u0995\u09c7 \u09ae\u09c2\u09b2\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 \u09b8\u09cd\u09aa\u09b0\u09cd\u09b6 \u0995\u09b0\u09b2\u09c7, \u0995\u09c7\u09a8\u09cd\u09a6\u09cd\u09b0 x<\/span>-\u0985\u0995\u09cd\u09b7\u09c7\u09b0 \u0989\u09aa\u09b0 \u0985\u09ac\u09b8\u09cd\u09a5\u09bf\u09a4 \u09b9\u09ac\u09c7 \u098f\u09ac\u0982 \u09ab\u09b2\u09c7 k=0<\/span> \u098f\u09ac\u0982 r=h, f^2=c=0<\/span>\u00a0(\u099a\u09bf\u09a4\u09cd\u09b0-\u09ee)\u00a0<\/span><\/li>\r\n
  8. (1) \u0993 (2) \u09ac\u09c3\u09a4\u09cd\u09a4\u09a6\u09cd\u09ac\u09df x<\/span>-\u0985\u0995\u09cd\u09b7\u0995\u09c7 \u09ae\u09c2\u09b2\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 \u09b8\u09cd\u09aa\u09b0\u09cd\u09b6 \u0995\u09b0\u09b2\u09c7, \u0995\u09c7\u09a8\u09cd\u09a6\u09cd\u09b0 y<\/span>-\u0985\u0995\u09cd\u09b7\u09c7\u09b0 \u0989\u09aa\u09b0 \u0985\u09ac\u09b8\u09cd\u09a5\u09bf\u09a4 \u09b9\u09ac\u09c7 \u098f\u09ac\u0982 \u09ab\u09b2\u09c7 h=0<\/span> \u098f\u09ac\u0982 r=k, g^2=c=0<\/span>\u00a0(\u099a\u09bf\u09a4\u09cd\u09b0-\u09ed)\u00a0<\/span><\/li>\r\n<\/ol>\r\n

     <\/p>\r\n\r\n\r\n\r\n

    \r\n
    \"\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1<\/figure>\r\n<\/div>\r\n\r\n\r\n\r\n

    \u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u09b8\u09ae\u09cd\u09aa\u09b0\u09cd\u0995\u09c7 \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09b0 \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09a8:<\/b><\/span><\/h2>\r\n\r\n\r\n\r\n
    \r\n
    \"\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1<\/figure>\r\n<\/div>\r\n\r\n\r\n\r\n

    O(-g,-f)<\/span> \u0995\u09c7\u09a8\u09cd\u09a6\u09cd\u09b0 \u098f\u09ac\u0982 r=\\sqrt{g^2+f^2-c}<\/span><\/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, x^2+y^2+2gx+2fy+c=0<\/span><\/span> \u098f\u09ac\u0982 \u09aa\u09cd\u09b0\u09a6\u09a4\u09cd\u09a4 \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 P(x_1,\u00a0y_1)<\/span><\/span>\u0964<\/span><\/p>\r\n\r\n\r\n\\therefore OP^2=(x_1+g)^2+(y_1+f)^2\\\\ \\Rightarrow OP^2=x_1^2+y_1^2+2gx_1+2fy_1+g^2+f^2\\\\ \\Rightarrow OP^2-r^2=x_1^2+y_1^2+2gx_1+2fy_1+g^2+f^2-(g^2+f^2-c)\\\\ \\Rightarrow OP2^-r^2=x_1^2+y_1^2+2gx_1+2fy_1+c\u00a0<\/span>\n\r\n\r\n\r\n

    P<\/span><\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u099f\u09bf \u09ac\u09c3\u09a4\u09cd\u09a4\u099f\u09bf\u09b0 \u09ac\u09be\u0987\u09b0\u09c7 \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09a8 \u0995\u09b0\u09b2\u09c7,<\/span>\u00a0<\/span><\/p>\r\n\r\n\r\nOP>r\\Rightarrow OP^2-r^2>0\\Rightarrow x_1^2+y_1^2+2gx_1+2fy_1+c>0\u00a0<\/span>\n\r\n\r\n\r\n

    P<\/span><\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u099f\u09bf \u09ac\u09c3\u09a4\u09cd\u09a4\u099f\u09bf\u09b0 \u0989\u09aa\u09b0\u09c7 \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09a8 \u0995\u09b0\u09b2\u09c7,<\/span>\u00a0<\/span><\/p>\r\n\r\n\r\nOP=r\\Rightarrow OP^2-r^2=0\\Rightarrow x_1^2+y_1^2+2gx_1+2fy_1+c=0 <\/span>\n\r\n\r\n\r\n

    P<\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u099f\u09bf \u09ac\u09c3\u09a4\u09cd\u09a4\u099f\u09bf\u09b0 \u09ad\u09bf\u09a4\u09b0\u09c7 \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09a8 \u0995\u09b0\u09b2\u09c7,<\/span>\u00a0<\/span><\/p>\r\n\r\n\r\nOP<r\\Rightarrow OP^2-r^2<0\\Rightarrow x_1^2+y_1^2+2gx_1+2fy_1+c<0\u00a0<\/span>\n\r\n\r\n\r\n

    \\therefore (x_1,y_1)<\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u099f\u09bf<\/span> x^2+y^2+2gx_1+2fy_1+c=0<\/span> \u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u09ac\u09be\u0987\u09b0\u09c7, \u0989\u09aa\u09b0\u09c7 \u09ac\u09be \u09ad\u09bf\u09a4\u09b0\u09c7 \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09a8 \u0995\u09b0\u09ac\u09c7 \u09af\u09a6\u09bf \u09af\u09a5\u09be\u0995\u09cd\u09b0\u09ae\u09c7 x^2+y^2+2gx_1+2fy_1+c>,=<\/span><\/span> \u09ac\u09be, <0<\/span><\/span>\u00a0\u09b9\u09df\u0964<\/span><\/p>\r\n\r\n\r\n\r\n

    \u09a6\u09c1\u0987\u099f\u09bf \u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u09aa\u09b0\u09b8\u09cd\u09aa\u09b0\u0995\u09c7 \u09b8\u09cd\u09aa\u09b0\u09cd\u09b6 \u0995\u09b0\u09be\u09b0 \u09b6\u09b0\u09cd\u09a4:<\/b><\/span><\/h2>\r\n\r\n\r\n\r\n

    \u09a6\u09c1\u0987\u099f\u09bf \u09ac\u09c3\u09a4\u09cd\u09a4 \u09aa\u09b0\u09b8\u09cd\u09aa\u09b0\u0995\u09c7 \u09b8\u09b0\u09cd\u09ac\u09cb\u099a\u09cd\u099a \u09a6\u09c1\u0987\u099f\u09bf \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 \u099b\u09c7\u09a6 \u0995\u09b0\u09c7\u0964 \u099b\u09c7\u09a6\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09a6\u09c1\u0987\u099f\u09bf \u09b8\u09ae\u09be\u09aa\u09a4\u09bf\u09a4 \u09b9\u09b2\u09c7 \u09ac\u09c3\u09a4\u09cd\u09a4 \u09a6\u09c1\u0987\u099f\u09bf \u09aa\u09b0\u09b8\u09cd\u09aa\u09b0\u0995\u09c7 \u09b8\u09cd\u09aa\u09b0\u09cd\u09b6 \u0995\u09b0\u09c7\u099b\u09c7 \u09ac\u09b2\u09be \u09b9\u09df\u0964<\/span>\u00a0<\/span><\/p>\r\n\r\n\r\n\r\n

    \u09a6\u09c1\u0987\u099f\u09bf \u09ac\u09c3\u09a4\u09cd\u09a4 \u09aa\u09b0\u09b8\u09cd\u09aa\u09b0\u0995\u09c7 \u09ac\u09b9\u09bf\u0983\u09b8\u09cd\u09a5\u09ad\u09be\u09ac\u09c7 \u0985\u09a5\u09ac\u09be \u0985\u09a8\u09cd\u09a4\u0983\u09b8\u09cd\u09a5\u09ad\u09be\u09ac\u09c7 \u09b8\u09cd\u09aa\u09b0\u09cd\u09b6 \u0995\u09b0\u09a4\u09c7 \u09aa\u09be\u09b0\u09c7\u0964\u00a0<\/span><\/p>\r\n\r\n\r\n\r\n

    \r\n
    \"\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1<\/figure>\r\n<\/div>\r\n\r\n\r\n\r\n

    A<\/span><\/span> \u0993 B<\/span><\/span> \u0995\u09c7\u09a8\u09cd\u09a6\u09cd\u09b0 \u098f\u09ac\u0982 r_1<\/span><\/span> \u0993 r_2<\/span><\/span> \u09ac\u09cd\u09af\u09be\u09b8\u09be\u09b0\u09cd\u09a7\u09ac\u09bf\u09b6\u09bf\u09b7\u09cd\u099f \u09ac\u09c3\u09a4\u09cd\u09a4 \u09a6\u09c1\u0987\u099f\u09bf \u09aa\u09b0\u09b8\u09cd\u09aa\u09b0\u0995\u09c7-<\/span><\/p>\r\n\r\n\r\n\r\n

      \r\n
    1. \u09ac\u09b9\u09bf\u0983\u09b8\u09cd\u09a5\u09ad\u09be\u09ac\u09c7 \u09b8\u09cd\u09aa\u09b0\u09cd\u09b6 \u0995\u09b0\u09ac\u09c7 \u09af\u09a6\u09bf \u098f\u09ac\u0982 \u0995\u09c7\u09ac\u09b2 \u09af\u09a6\u09bf AB=r_1+r_2<\/span><\/span> \u098f\u09ac\u0982<\/span><\/li>\r\n
    2. \u0985\u09a8\u09cd\u09a4\u0983\u09b8\u09cd\u09a5\u09ad\u09be\u09ac\u09c7 \u09b8\u09cd\u09aa\u09b0\u09cd\u09b6 \u0995\u09b0\u09ac\u09c7 \u09af\u09a6\u09bf \u098f\u09ac\u0982 \u0995\u09c7\u09ac\u09b2 \u09af\u09a6\u09bf AB=r_1\\sim r_2<\/span>\u00a0\u09b9\u09df\u0964<\/li>\r\n<\/ol>\r\n

       <\/p>\r\n

      \r\n
      \r\n
      \r\n
      \r\n

      \u00a0<\/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<\/div>\r\n<\/div>\r\n<\/div>\r\n