{"id":3924,"date":"2021-01-30T12:29:20","date_gmt":"2021-01-30T12:29:20","guid":{"rendered":"https:\/\/stage-wp.10minuteschool.com\/?p=3924"},"modified":"2023-06-22T17:16:16","modified_gmt":"2023-06-22T11:16:16","slug":"general-equation-of-conic","status":"publish","type":"post","link":"https:\/\/10minuteschool.com\/content\/general-equation-of-conic\/","title":{"rendered":"\u0995\u09a3\u09bf\u0995\u09c7\u09b0 \u09b8\u09be\u09a7\u09be\u09b0\u09a3 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u098f\u09ac\u0982 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u09b9\u09a4\u09c7 \u0995\u09a3\u09bf\u0995 \u099a\u09bf\u09b9\u09cd\u09a8\u09bf\u09a4\u0995\u09b0\u09a3 (General equation of Conic Explanation and Conic identification)"},"content":{"rendered":"

\u0995\u09a3\u09bf\u0995\u09c7\u09b0 \u09b8\u09be\u09a7\u09be\u09b0\u09a3 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 <\/b>(General equation of Conic)<\/b><\/h2>\n

\"Graph\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, \u0995\u09cb\u09a8\u09cb \u09b8\u09ae\u09a4\u09b2\u09c7 \u098f\u0995\u099f\u09bf \u0995\u09a3\u09bf\u0995\u09c7\u09b0 \u09ab\u09cb\u0995\u09be\u09b8 (\u09b8\u09cd\u09a5\u09bf\u09b0 \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1) <\/span>S (\u03b1, \u03b2)<\/span> \u098f\u09ac\u0982 \u09a8\u09bf\u09df\u09be\u09ae\u0995 \u09b0\u09c7\u0996\u09be (\u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be) A B \\equiv L x+m y+n=0<\/span><\/span>\u00a0\u0995\u09a3\u09bf\u0995\u099f\u09bf\u09b0 \u0989\u09aa\u09b0\u09b8\u09cd\u09a5 \u09af\u09c7\u0995\u09c7\u09be\u09a8\u09c7\u09be \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 <\/span>P (x, y)<\/span> \u098f\u09ac\u0982 <\/span>PM \u22a5 AB<\/span> \u09b9\u09b2\u09c7 \u0995\u09a3\u09bf\u0995\u09c7\u09b0 \u09b8\u0982\u099c\u09cd\u099e\u09be \u09b9\u09a4\u09c7 \u09aa\u09be\u0987,<\/span><\/p>\n

SPPM=e (\u0989\u09ce\u0995\u09c7\u09a8\u09cd\u09a6\u09cd\u09b0\u09bf\u0995\u09a4\u09be)<\/strong><\/p>\n\\frac{S P}{P M}=e<\/span>\n\\Rightarrow S P=e P M<\/span>\n

\\Rightarrow \\sqrt{(x-\\alpha)^{2}+(y-\\beta)^{2}}=e \\cdot \\frac{l x+m y+n}{\\sqrt{l^{2}+m^{2}}}<\/span> \u09af\u09be \u0995\u09a3\u09bf\u0995\u09c7\u09b0 \u09b8\u09be\u09a7\u09be\u09b0\u09a3 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3<\/span><\/p>\n

    \n
  • e=1<\/span> \u09b9\u09b2\u09c7, \u0995\u09a3\u09bf\u0995\u099f\u09bf \u09aa\u09b0\u09be\u09ac\u09c3\u09a4\u09cd\u09a4 \u098f\u09ac\u0982 \u098f\u09b0 \u09b8\u09be\u09a7\u09be\u09b0\u09a3 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3: (x-\\alpha)^{2}+(y-\\beta)^{2}=\\frac{(l x+m y+n)^{2}}{l^{2}+m^{2}}<\/span><\/span> \u098f\u09ac\u0982 \u0986\u09a6\u09b0\u09cd\u09b6 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3: y^{2}=4 a x<\/span><\/span><\/li>\n<\/ul>\n

    \"Graph<\/p>\n

      \n
    • 0 < e < 1<\/span> \u09b9\u09b2\u09c7, \u0995\u09a3\u09bf\u0995\u099f\u09bf \u0989\u09aa\u09ac\u09c3\u09a4\u09cd\u09a4 \u098f\u09ac\u0982 \u098f\u09b0 \u09b8\u09be\u09a7\u09be\u09b0\u09a3 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3: (x-\\alpha)^{2}+(y-\\beta)^{2}=e^{2} \\frac{(l x+m y+n)^{2}}{l^{2}+m^{2}}<\/span><\/span> \u098f\u09ac\u0982 \u0986\u09a6\u09b0\u09cd\u09b6 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3: \\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1<\/span><\/span><\/li>\n<\/ul>\n

      e = 0<\/span> \u09b9\u09b2\u09c7 \u0989\u09aa\u09ac\u09c3\u09a4\u09cd\u09a4\u099f\u09bf \u09ac\u09c3\u09a4\u09cd\u09a4\u09c7 \u09aa\u09b0\u09bf\u09a3\u09a4 \u09b9\u09ac\u09c7\u0964<\/span><\/p>\n

      \"Graph<\/p>\n

        \n
      • e > 1<\/span> \u09b9\u09b2\u09c7, \u0995\u09a3\u09bf\u0995\u099f\u09bf \u09b9\u09ac\u09c7 \u0985\u09a7\u09bf\u09ac\u09c3\u09a4\u09cd\u09a4 \u098f\u09ac\u0982 \u098f\u09b0 \u09b8\u09be\u09a7\u09be\u09b0\u09a3 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3: (x-\\alpha)^{2}+(y-\\beta)^{2}=e^{2} \\frac{(l x+m y+n)^{2}}{l^{2}+m^{2}}<\/span><\/span> \u098f\u09ac\u0982 \u0986\u09a6\u09b0\u09cd\u09b6 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3: \\frac{x^{2}}{a^{2}}-\\frac{y^{2}}{b^{2}}=1<\/span><\/span><\/li>\n<\/ul>\n

        \"Graph<\/p>\n

        \u09b8\u09be\u09a7\u09be\u09b0\u09a3 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u09b9\u09a4\u09c7 \u0995\u09a3\u09bf\u0995 \u099a\u09bf\u09b9\u09cd\u09a8\u09bf\u09a4\u0995\u09b0\u09a3<\/b>(Conic identification from general equation)<\/b><\/h2>\na x^{2}+2 h x y+b y^{2}+2 g x+2 f y+c=0<\/span>\n

        Case-1:<\/b>\u00a0<\/span><\/p>\n

        A=\\left|\\begin{array}{lll}\n\na & h & g \\\\\n\nh & b & f \\\\\n\ng & f & c\n\n\\end{array}\\right| ; A=0<\/span> \u09b9\u09b2\u09c7, \u098f\u0995\u099c\u09cb\u09dc\u09be \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be \u09a8\u09bf\u09b0\u09cd\u09a6\u09c7\u09b6 \u0995\u09b0\u09c7\u0964<\/span><\/p>\n

        Case-2:\u00a0<\/b><\/p>\n

        A \\neq 0 ; a=b ; h=0 ; g^{2}+f^{2}-c \\geq 0 \\text {\u09b9\u09b2\u09c7, } x^{2}+y^{2}+2 g x+2 f y+c=0<\/span>\u00a0 \u09af\u09be \u098f\u0995\u099f\u09bf \u09ac\u09c3\u09a4\u09cd\u09a4 \u09a8\u09bf\u09b0\u09cd\u09a6\u09c7\u09b6 \u0995\u09b0\u09c7\u0964<\/span><\/p>\n