\u300c\u6955\u5186\u3068\u306f\uff0c2\u5b9a\u70b9\u304b\u3089\u306e\u8ddd\u96e2\u306e\u548c\u304c\u4e00\u5b9a\u3068\u306a\u308b\u3088\u3046\u306a\u5e73\u9762\u4e0a\u306e\u70b9\u306e\u8ecc\u8de1\u3067\u3042\u308b\u3002\u300d\u3053\u306e\u5b9a\u70b9\u306e\u3053\u3068\u3092\uff08\u7b11\u70b9\u3067\u306f\u306a\u304f\uff09\u7126\u70b9\u3068\u3044\u3046\u3002<\/p>\n
\u3053\u306e\u5b9a\u7fa9\u304b\u3089\uff0c\u6955\u5186\u306e\u65b9\u7a0b\u5f0f\u304c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8868\u3055\u308c\u308b\u3053\u3068\u3092\u793a\u3057\u3066\u304a\u304f\u3002<\/p>\n
\u6955\u5186\u306e\u4e2d\u5fc3\u3092\u539f\u70b9\u3068\u3057\u305f\u30c7\u30ab\u30eb\u30c8\u5ea7\u6a19 \\(X, Y\\) \u3067\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8868\u3055\u308c\u308b\u3053\u3068\u3092\u793a\u3059\u3002<\/p>\n
$$\\frac{X^2}{a^2} + \\frac{Y^2}{b^2} = 1$$<\/p>\n
\u307e\u305a\uff0c2\u3064\u306e\u7126\u70b9\u3092 \\(X\\) \u8ef8\u4e0a\u306b\u304a\u304d\uff0c\\(F(c, 0), \\ F'(-c, 0)\\) \u3068\u3059\u308b\u3002\uff08\\(c > 0\\)\uff09\u70b9 \\(P\\) \u306e\u5ea7\u6a19\u3092 \\((X, Y)\\) \u3068\u3059\u308b\u3068\uff0c\u984c\u610f\u304b\u3089<\/p>\n
\\begin{eqnarray}
\nPF + PF’ &=& \\mbox{const.} \\equiv 2 a \\\\
\n\\sqrt{(X-c)^2 + Y^2} + \\sqrt{(X+c)^2 + Y^2} &=& 2a \\\\
\n(X+c)^2 + Y^2 &=& \\left\\{2 a –\u00a0 \\sqrt{(X+c)^2 + Y^2}\\right\\}^2\\\\
\n&=& 4a^2 – 4a\\sqrt{(X+c)^2 + Y^2} + (X+c)^2 + Y^2\\\\
\n\\therefore\\ \\ \\sqrt{(X-c)^2 + Y^2} &=& a – \\frac{c}{a} X \\\\
\n(X-c)^2 + Y^2 &=& a^2 – 2 c X + \\frac{c^2}{a^2} X^2\\\\
\n\\left(1 – \\frac{c^2}{a^2} \\right) X^2 + Y^2 &=& a^2 – c^2\\\\
\n\\therefore\\ \\ \\frac{X^2}{a^2} + \\frac{Y^2}{a^2 – c^2} &=& 1
\n\\end{eqnarray}<\/p>\n
\u3053\u3053\u3067\uff0c\\(b^2 \\equiv a^2 – c^2\\) \u3068\u304a\u3051\u3070<\/p>\n
$$\\frac{X^2}{a^2} + \\frac{Y^2}{b^2} = 1$$<\/p>\n
\u306a\u304a\uff0c\\(a > c\\) \u3067\u3042\u308b\u3053\u3068\u3092\u4eee\u5b9a\u3057\u3066 \\(b^2 \\equiv a^2 – c^2\\)\u00a0 \u3068\u304a\u3044\u305f\u304c\uff0c\u3053\u306e\u72b6\u6cc1\u3067\u5c0e\u304b\u308c\u305f\u6955\u5186\u306e\u65b9\u7a0b\u5f0f\u306e\u6a19\u6e96\u5f62\u304b\u3089\u4ee5\u4e0b\u306e\u3053\u3068\u304c\u308f\u304b\u308b\u3002<\/p>\n
$$ -a \\leq X \\leq a, \\quad -b \\leq Y \\leq b$$<\/p>\n
\u3064\u307e\u308a\uff0c\\(a > c\\) \u3059\u306a\u308f\u3061 \\(0 \\leq e < 1\\) \u3068\u3044\u3046\u4eee\u5b9a\u306f\uff0c\u904b\u52d5\u304c\u6709\u754c\u3067\u3042\u308b\u3053\u3068\uff0c\u8a00\u3044\u63db\u3048\u308c\u3070\u904b\u52d5\u304c\u675f\u7e1b\u72b6\u614b\u306b\u3042\u308b\u3053\u3068\u3092\u4eee\u5b9a\u3057\u3066\u3044\u308b\u3053\u3068\u306b\u306a\u3063\u3066\u3044\u308b\u3002<\/p>\n
$$\\sqrt{(X-c)^2 + Y^2} + \\sqrt{(X+c)^2 + Y^2} = 2a$$\u304c\\(X = 0\\) \u306e\u3068\u304d\uff0c\\(Y \\neq 0\\) \u3067\u3042\u308b\u3068\u3044\u3046\u6761\u4ef6\u304b\u3089<\/p>\n
\\begin{eqnarray}
\n2 \\sqrt{c^2 + Y^2} &=& 2 a \\\\
\n\\therefore\\ \\ c < a
\n\\end{eqnarray}<\/p>\n
\\( a > b\\) \u306e\u5834\u5408\u306b \\(a\\) \u3092\u9577\u534a\u5f84<\/strong><\/span>\uff0c\\(b\\) \u3092\u77ed\u534a\u5f84<\/strong><\/span>\u3068\u3044\u3046\u3002<\/p>\n \u307e\u305f\uff0c\u7126\u70b9\u306e\u5ea7\u6a19\u306b\u3042\u3089\u308f\u308c\u308b\u5b9a\u6570 \\(c\\) \u3092\u4e2d\u5fc3\u304b\u3089\uff08\u9577\u534a\u5f84\u3068\u306e\u6bd4\u3067\uff09\u3069\u306e\u7a0b\u5ea6\u305a\u308c\u3066\u3044\u308b\u304b\u3092\u8868\u3059\u96e2\u5fc3\u7387<\/strong><\/span> \\(e\\) \u3092\u5b9a\u7fa9\u3057\u3066\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u304f\u3002<\/p>\n $$c = a e, \\quad 0 \\leq e < 1$$<\/p>\n \u3059\u308b\u3068\uff0c\u77ed\u534a\u5f84 \\(b\\) \u306f<\/p>\n $$b = \\sqrt{a^2 – c^2} = a \\sqrt{1 – e^2}$$\u306e\u3088\u3046\u306b\u3082\u66f8\u3051\u308b\u3002<\/p>\n \u30b1\u30d7\u30e9\u30fc\u306e\u7b2c1\u6cd5\u5247\u306b\u3044\u307f\u3058\u304f\u3082\u8868\u3055\u308c\u3066\u3044\u308b\u3088\u3046\u306b\uff0c\u5929\u6587\u5b66\u3067\u51fa\u3066\u304f\u308b\u6955\u5186\u306e\u5f0f\u3068\u3044\u3048\u3070\uff0c\u3053\u3063\u3061\u3002<\/p>\n \u6955\u5186\u306e\u7126\u70b9 \\(F\\) \u3092\u539f\u70b9\u3068\u3057\u305f\u6975\u5ea7\u6a19\u3092 \\(r, \\phi\\) \u3068\u3059\u308b\u3068\uff0c<\/p>\n \\begin{eqnarray} \u3053\u308c\u3092\u6955\u5186\u306e\u65b9\u7a0b\u5f0f\u306e\u6a19\u6e96\u5f62\u306b\u4ee3\u5165\u3057\u3066<\/p>\n \\begin{eqnarray} \u5c55\u958b\u3059\u308b\u3068 \\(r\\) \u306b\u3064\u3044\u30662\u6b21\u65b9\u7a0b\u5f0f\u3068\u306a\u308a\uff0c\u56e0\u6570\u5206\u89e3\u3067\u304d\u3066<\/p>\n \\begin{eqnarray} \\(r > 0\\) \u3067\u3042\u308b\u304b\u3089\u89e3\u306f<\/p>\n $$r = \\frac{a(1-e^2)}{1+e\\cos\\phi}$$<\/p>\n \u306a\u306e\u3067\uff0c\u4eca\u5f8c\u306f\u3053\u306e\u5f0f\u3092\u898b\u305f\u3089\uff0c\u3042\u3063\u3053\u308c\u306f\u9577\u534a\u5f84 \\(a\\)\uff0c\u96e2\u5fc3\u7387 \\(e\\) \u306e\u6955\u5186\u3060\uff01\u3068\u810a\u9ac4\u53cd\u5c04\u3059\u308b\u3088\u3046\u306b\u3002<\/p>\n \u306a\u304a\uff0c\u5b66\u751f\u3055\u3093\u306e\u89e3\u7b54\u3092\u307f\u3066\u3044\u308b\u3068\uff0c\\(r\\) \u306e2\u6b21\u65b9\u7a0b\u5f0f\u306b\u3059\u308b\u4e0a\u8a18\u306e\u3084\u308a\u65b9\u3060\u3068\u56e0\u6570\u5206\u89e3\u304c\u3067\u304d\u306a\u304b\u3063\u305f\u308a\uff0c\u89e3\u306e\u516c\u5f0f\u3092\u4f7f\u304a\u3046\u3068\u3057\u3066\u8a08\u7b97\u30df\u30b9\u3092\u3057\u3066\u3057\u307e\u3046\u30b1\u30fc\u30b9\u3082\u591a\u3005\u898b\u304b\u3051\u308b\u3002\u5225\u89e3\u3068\u3057\u3066\u4e09\u89d2\u5f62 \\(FPF’\\) \u306b\u5bfe\u3059\u308b\u4f59\u5f26\u5b9a\u7406\u3092\u4f7f\u3046\u4f8b\u3002<\/p>\n \u7126\u70b9 \\(F\\) \u3092\u539f\u70b9\u3068\u3057\u305f\u6975\u5ea7\u6a19\u3092 \\(r, \\phi\\) \u3068\u3057\u3066\u3044\u308b\u306e\u3067\uff0c\\(FP = r\\)\uff0c\\(FF’ = 2 ae\\)\u3002\u307e\u305f\uff0c\\(PF’ = r’\\) \u3068\u3059\u308b\u3068\uff0c\u4f59\u5f26\u5b9a\u7406\u3068\u6955\u5186\u306e\u5b9a\u7fa9\u304b\u3089<\/p>\n \\begin{eqnarray} \u3068\u306a\u308a\uff0c$$r = \\frac{a(1-e^2)}{1+e\\cos\\phi}$$<\/p>\n","protected":false},"excerpt":{"rendered":" \u6955\u5186\u306e\u5b9a\u7fa9 <\/p>\n \u300c\u6955\u5186\u3068\u306f\uff0c2\u5b9a\u70b9\u304b\u3089\u306e\u8ddd\u96e2\u306e\u548c\u304c\u4e00\u5b9a\u3068\u306a\u308b\u3088\u3046\u306a\u5e73\u9762\u4e0a\u306e\u70b9\u306e\u8ecc\u8de1\u3067\u3042\u308b\u3002\u300d\u3053\u306e\u5b9a\u70b9\u306e\u3053\u3068\u3092\uff08\u7b11\u70b9\u3067\u306f\u306a\u304f\uff09\u7126\u70b9\u3068\u3044\u3046\u3002<\/p>\n \u3053\u306e\u5b9a\u7fa9\u304b\u3089\uff0c\u6955\u5186\u306e\u65b9\u7a0b\u5f0f\u304c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8868\u3055\u308c\u308b\u3053\u3068\u3092\u793a\u3057\u3066\u304a\u304f\u3002<\/p>\u6955\u5186\u306e\u7126\u70b9\u306e1\u3064\u3092\u539f\u70b9\u3068\u3057\u305f\u6975\u5ea7\u6a19\u3067\u8868\u3057\u305f\u6955\u5186\u306e\u65b9\u7a0b\u5f0f<\/h3>\n
\nX &=& c + r \\cos\\phi = a e + r \\cos\\phi \\\\
\nY &=& r \\sin\\phi
\n\\end{eqnarray}<\/p>\n
\n\\frac{(ae+r\\cos\\phi)^2}{a^2} + \\frac{(r \\sin\\phi)^2}{a^2 (1 – e^2)} &=& 1
\n\\end{eqnarray}<\/p>\n
\n\\left\\{ (1+e\\cos\\phi) r – a(1-e^2)\\right\\} \\left\\{ (1-e\\cos\\phi) r + a(1-e^2)\\right\\} &=& 0
\n\\end{eqnarray}<\/p>\n\u5225\u89e3<\/h4>\n
\n(r’)^2 &=& r^2 + (2ae)^2 – 2 r \\times (2ae) \\times \\cos\\left( \\pi – \\phi\\right)\\\\
\nr’ + r &=& 2 a \\\\
\n\\therefore\\ \\ (2a)^2 – 4 a r + r^2 &=& r^2 + (2ae)^2 + 2 r \\times (2ae) \\times \\cos \\phi \\\\
\nr \\left(1 + e \\cos\\phi \\right) &=& a (1 – e^2)
\n\\end{eqnarray}<\/p>\n