{"id":3943,"date":"2021-01-30T12:24:37","date_gmt":"2021-01-30T12:24:37","guid":{"rendered":"https:\/\/stage-wp.10minuteschool.com\/?p=3943"},"modified":"2023-06-22T17:23:02","modified_gmt":"2023-06-22T11:23:02","slug":"parabola","status":"publish","type":"post","link":"https:\/\/10minuteschool.com\/content\/parabola\/","title":{"rendered":"\u09aa\u09b0\u09be\u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u09aa\u09b0\u09bf\u099a\u09bf\u09a4\u09bf (Introduction to the parabola)"},"content":{"rendered":"

\u09aa\u09b0\u09be\u09ac\u09c3\u09a4\u09cd\u09a4 (<\/b>Parabola)<\/b><\/h2>\n

\"Graph<\/p>\n\n\n\n\n\n\n\n
<\/td>\ny^{2}=4 a x<\/span>\u00a0<\/b><\/p>\n

\u00a0[<\/b>a>0<\/span>]\u00a0<\/b><\/td>\n

x^{2}=4 a y<\/span>\u00a0<\/b><\/p>\n

\u00a0[<\/b>a>0<\/span>]\u00a0<\/b><\/td>\n

(y-\\beta)^{2}=4 a(x-\\alpha)<\/span>\u00a0 [<\/b>a>0]<\/span><\/td>\n(x-\\alpha)^{2}=4 a(y-\\beta)<\/span>\u00a0 [<\/b>a>0<\/span>]<\/b><\/td>\n<\/tr>\n
\u099a\u09bf\u09a4\u09cd\u09b0<\/b><\/td>\n\"<\/td>\n\"x2=4ay\u00a0<\/td>\n\"(y-)2=4a(x-)\u00a0<\/td>\n\"(x-)2=4a(y-)\u00a0<\/td>\n<\/tr>\n
\u09b6\u09c0\u09b0\u09cd\u09b7\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09b0 \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995 <\/b>(A)<\/span><\/td>\n(0,\u00a00)<\/span><\/td>\n(0,\u00a00)<\/span><\/td>\n(\\alpha, \\beta)<\/span><\/td>\n(\\alpha, \\beta)<\/span><\/td>\n<\/tr>\n
\u09ab\u09cb\u0995\u09be\u09b8\/ \u0989\u09aa\u0995\u09c7\u09a8\u09cd\u09a6\u09cd\u09b0\u09c7\u09b0 \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995<\/b>\u00a0<\/b>(S)<\/span><\/td>\n(a,\u00a00)<\/span><\/td>\n(0,\u00a0a)<\/span><\/td>\n(a+\\alpha, \\beta)<\/span><\/td>\n(\\alpha, a+\\beta)<\/span><\/td>\n<\/tr>\n
\u09a6\u09bf\u0995\u09be\u0995\u09cd\u09b7\/ \u09a8\u09bf\u09df\u09be\u09ae\u0995\u09b0\u09c7\u0996\u09be\u09b0 \u09aa\u09be\u09a6\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09b0 \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995<\/b><\/td>\n(-a,\u00a00)<\/span><\/td>\n(0,\u00a0-a)<\/span><\/td>\n(-a+\\alpha, \\beta)<\/span><\/td>\n (\\alpha,-a+\\beta)<\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

 <\/p>\n\n\n\n\n\n\n\n\n\n\n\n\n
\u09a6\u09bf\u0995\u09be\u0995\u09cd\u09b7\/<\/b>\u00a0<\/b>\u09a8\u09bf\u09df\u09be\u09ae\u0995\u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3<\/b><\/td>\n <\/p>\n

x+a=0<\/p>\n

\\Rightarrow<\/span>x = a – 1<\/p>\n

 <\/td>\n

?+?=?
\n\u21d2?=\u2212?<\/td>\n		?\u2212?+?=?<\/td>\n?\u2212?+?=?<\/td>\n<\/tr>\n
\u0985\u0995\u09cd\u09b7\u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3<\/span><\/td>\ny=0<\/span><\/td>\nx=0<\/span><\/td>\n\\begin{array}{c}\n\ny=\\beta \\\\\n\n\\Rightarrow y-\\beta=0\n\n\\end{array}<\/span><\/td>\n\\begin{array}{c}\n\nx=\\alpha \\\\\n\n\\Rightarrow x-\\alpha=0\n\n\\end{array}<\/span><\/td>\n<\/tr>\n
\u0989\u09aa\u0995\u09c7\u09a8\u09cd\u09a6\u09cd\u09b0\u09bf\u0995 \u09b2\u09ae\u09cd\u09ac\u09c7\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3<\/span><\/td>\nx=a<\/span><\/td>\ny=a<\/span><\/td>\n\\begin{array}{c}\n\nx-\\alpha-a=0 \\\\\n\n\\Rightarrow x-\\alpha=a\n\n\\end{array}<\/span><\/td>\n \\begin{array}{c}\n\ny-\\beta-a=0 \\\\\n\n\\Rightarrow y-\\beta=a\n\n\\end{array}<\/span><\/span><\/td>\n<\/tr>\n
\u0989\u09aa\u0995\u09c7\u09a8\u09cd\u09a6\u09cd\u09b0\u09bf\u0995 \u09b2\u09ae\u09cd\u09ac\u09c7\u09b0 \u09a6\u09c8\u09b0\u09cd\u0998\u09cd\u09af<\/span><\/td>\n4 |a|<\/span><\/td>\n4 |a|<\/span><\/td>\n4 |a|<\/span><\/td>\n4 |a|<\/span><\/td>\n<\/tr>\n
\u0989\u09aa\u0995\u09c7\u09a8\u09cd\u09a6\u09cd\u09b0\u09bf\u0995 \u09b2\u09ae\u09cd\u09ac\u09c7\u09b0 \u09aa\u09cd\u09b0\u09be\u09a8\u09cd\u09a4\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a6\u09cd\u09ac\u09df<\/span><\/td>\n(a,\u00a0\u00b12a)<\/span><\/td>\n(\u00b12a,\u00a0a)<\/span><\/td>\n(a+\\alpha, \\pm 2 a+\\beta)<\/span><\/td>\n(\\pm 2 a+\\alpha, a+\\beta)<\/span><\/td>\n<\/tr>\n
\u09ab\u09cb\u0995\u09be\u09b8 \u09a6\u09c2\u09b0\u09a4\u09cd\u09ac\/ \u0989\u09aa\u0995\u09c7\u09a8\u09cd\u09a6\u09cd\u09b0\u09bf\u0995 \u09a6\u09c2\u09b0\u09a4\u09cd\u09ac<\/span><\/td>\nSP=x+a<\/span><\/td>\nSP=y+a<\/span><\/td>\nx-<\/span>+a<\/span><\/td>\ny-<\/span>+a<\/span><\/td>\n<\/tr>\n
\u09b6\u09c0\u09b0\u09cd\u09b7\u09c7 \u09b8\u09cd\u09aa\u09b0\u09cd\u09b6\u0995\u09c7\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3<\/span><\/td>\nx=0<\/span><\/td>\ny=0<\/span><\/td>\nx=\u03b1<\/span><\/p>\n

\u21d2x-\u03b1=0<\/span><\/td>\n

y=\u03b2<\/span><\/p>\n

\u21d2y-\u03b2=0<\/span><\/td>\n<\/tr>\n

\u0989\u09aa\u0995\u09c7\u09a8\u09cd\u09a6\u09cd\u09b0 \u0993 \u09b6\u09c0\u09b0\u09cd\u09b7\u09c7\u09b0 \u09a6\u09c2\u09b0\u09a4\u09cd\u09ac<\/span><\/td>\na<\/span><\/td>\na<\/span><\/td>\n<\/td>\n<\/td>\n<\/tr>\n
\u09aa\u09b0\u09be\u09ae\u09bf\u09a4\u09bf\u0995 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3<\/span><\/td>\n \\begin{array}{c}\n\nx=a t^{2} \\\\\n\ny=2 a t\n\n\\end{array}<\/span><\/span><\/td>\n\\begin{array}{l}\n\ny=a t^{2} \\\\\n\nx=2 a t\n\n\\end{array}<\/span><\/td>\n<\/td>\n<\/td>\n<\/tr>\n
\u09aa\u09cb\u09b2\u09be\u09b0 \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995\u09c7 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3<\/span><\/td>\nr=<\/span><\/p>\n

4acot\u03b8cosec\u03b8<\/span><\/td>\n

r=<\/span><\/p>\n

4atan\u03b8sec\u03b8<\/span><\/td>\n

<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

 <\/p>\n

    \n
  • x<\/span> \u0985\u0995\u09cd\u09b7\u09c7\u09b0 \u09b8\u09ae\u09be\u09a8\u09cd\u09a4\u09b0\u09be\u09b2 \u0985\u0995\u09cd\u09b7\u09ac\u09bf\u09b6\u09bf\u09b7\u09cd\u099f<\/span>\u00a0<\/span>\u09aa\u09b0\u09be\u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3:x=a y^{2}+b y+c<\/span><\/span><\/li>\n<\/ul>\n

    \u09b6\u09c0\u09b0\u09cd\u09b7 <\/span>\\left(-\\frac{b^{2}-4 a c}{4 a},-\\frac{b}{2 a}\\right)<\/span><\/span>\u00a0\u0989\u09aa\u0995\u09c7\u09a8\u09cd\u09a6\u09cd\u09b0\u09bf\u0995 \u09b2\u09ae\u09cd\u09ac <\/span>=\\frac{1}{|a|}<\/span><\/span><\/p>\n

      \n
    • y<\/span> \u0985\u0995\u09cd\u09b7\u09c7\u09b0 \u09b8\u09ae\u09be\u09a8\u09cd\u09a4\u09b0\u09be\u09b2 \u0985\u0995\u09cd\u09b7\u09ac\u09bf\u09b6\u09bf\u09b7\u09cd\u099f<\/span>\u00a0<\/span>\u09aa\u09b0\u09be\u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3: y=a x^{2}+b x+c<\/span><\/span><\/li>\n<\/ul>\n

      \u09b6\u09c0\u09b0\u09cd\u09b7\u00a0 <\/span>\\left(-\\frac{b}{2 a},-\\frac{b^{2}-4 a c}{4 a}\\right)<\/span><\/span> \u0989\u09aa\u0995\u09c7\u09a8\u09cd\u09a6\u09cd\u09b0\u09bf\u0995 \u09b2\u09ae\u09cd\u09ac =\\frac{1}{|a|}<\/span><\/span><\/p>\n

        \n
      • Y^{2}=4 a x<\/span><\/span>\u00a0\u09aa\u09b0\u09be\u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \\left(x_{1}, y_{1}\\right)<\/span><\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 \u09b8\u09cd\u09aa\u09b0\u09cd\u09b6\u0995\u09c7\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3: y y_{1}=2 a\\left(x+x_{1}\\right)<\/span><\/span><\/li>\n
      • e=0<\/strong> \u09b9\u09b2\u09c7 \u098f\u0995\u099f\u09bf \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09ac\u09c3\u09a4\u09cd\u09a4 \u09aa\u09be\u0993\u09df\u09be \u09af\u09be\u09df\u0964<\/span>\u00a0<\/span><\/li>\n<\/ul>\n

        [<\/span>e<\/span> \u098f\u09b0 \u0986\u09b0 \u098f\u09ae\u09a8 \u0995\u09cb\u09a8\u09cb \u09ae\u09be\u09a8 \u09a8\u09be\u0987 \u09af\u09c7, \u0995\u09a3\u09bf\u0995\u09c7\u09b0 \u09b8\u0982\u099c\u09cd\u099e\u09be \u09b9\u09a4\u09c7 \u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u09aa\u09be\u0993\u09df\u09be \u09af\u09be\u09df\u0964 \u098f\u0987 \u0995\u09be\u09b0\u09a3\u09c7 \u09ac\u09c3\u09a4\u09cd\u09a4\u0995\u09c7 \u0995\u09a3\u09bf\u0995 \u09ac\u09b2\u09be \u09af\u09be\u09df \u09a8\u09be\u0964]<\/span><\/p>\n