\u76f4\u7dda \\(\\ell : \\ mx+ny = 1\\) \u304c, \u6955\u5186 \\(C : \\ \\dfrac{x^2}{a^2} +\\dfrac{y^2}{b^2} =1 \\ ( a \\gt b \\gt 0 )\\) \u306b\u63a5\u3057\u306a\u304c\u3089\u52d5\u304f\u3068\u3059\u308b.<\/p>\r\n
(1)<\/strong>\u3000\u70b9 \\((m,n)\\) \u306e\u8ecc\u8de1\u306f\u6955\u5186\u306b\u306a\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n (2)<\/strong>\u3000\\(C\\) \u306e\u7126\u70b9 \\(F _ 1 \\left( -\\sqrt{a^2-b^2} , 0 \\right)\\) \u3068 \\(\\ell\\) \u3068\u306e\u8ddd\u96e2\u3092 \\(d _ 1\\) \u3068\u3057, \u3082\u3046 \\(1\\) \u3064\u306e\u7126\u70b9 \\(F _ 2 \\left( \\sqrt{a^2-b^2} , 0 \\right)\\) \u3068 \\(\\ell\\) \u3068\u306e\u8ddd\u96e2\u3092 \\(d _ 2\\) \u3068\u3059\u308b. \u3053\u306e\u3068\u304d \\(d _ 1 d _ 2 = b^2\\) \u3092\u793a\u305b.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \\(C\\) \u4e0a\u306e\u70b9 \\(( a \\cos \\theta , b \\sin \\theta) \\ ( 0 \\leqq \\theta \\lt 2\\pi )\\) \u306b\u304a\u3051\u308b\u63a5\u7dda\u306e\u5f0f\u306f\r\n\\[\\begin{align}\r\n\\dfrac{x (a \\cos \\theta)}{a^2} +\\dfrac{y (b \\sin \\theta)}{b^2} & = 1 \\\\\r\n\\dfrac{x \\cos \\theta}{a} +\\dfrac{y \\sin \\theta}{b} & = 1\r\n\\end{align}\\]\r\n\u3053\u308c\u3068 \\(\\ell\\) \u306e\u5f0f\u3092\u6bd4\u8f03\u3059\u308c\u3070\r\n\\[\r\nm = \\dfrac{\\cos \\theta}{a} , \\ n =\\dfrac{\\sin \\theta}{b}\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \u70b9 \\((m,n)\\) \u306e\u8ecc\u8de1\u306f\r\n\\[\r\n\\underline{\\text{\u6955\u5186} : \\ a^2x^2+b^2y^2 = 1}\r\n\\]\r\n (2)<\/strong><\/p>\r\n[1] \u3092\u5909\u5f62\u3059\u308b\u3068\r\n\\[\r\nx b \\cos \\theta +y a \\sin \\theta -ab = 0\r\n\\]\r\n\u3053\u308c\u3092\u7528\u3044\u308b\u3068\r\n\\[\\begin{align}\r\nd _ 1 & = \\dfrac{\\left| -b \\sqrt{a^2-b^2} \\cos \\theta -ab \\right|}{\\sqrt{b^2 \\cos^2 \\theta + a^2 \\sin^2 \\theta}} \\\\\r\n& = b \\cdot \\dfrac{a +\\sqrt{a^2-b^2} \\cos \\theta}{\\sqrt{a^2 -(a^2-b^2) \\cos^2 \\theta}} , \\\\\r\nd _ 2 & = \\dfrac{\\left| b \\sqrt{a^2-b^2} \\cos \\theta -ab \\right|}{\\sqrt{b^2 \\cos^2 \\theta + a^2 \\sin^2 \\theta}} \\\\\r\n& = b \\cdot \\dfrac{a -\\sqrt{a^2-b^2} \\cos \\theta}{\\sqrt{a^2 -(a^2-b^2) \\cos^2 \\theta}}\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\r\nd _ 1 d _ 2 = b^2 \\cdot \\dfrac{a^2 -(a^2-b^2) \\cos^2 \\theta}{a^2 -(a^2-b^2) \\cos^2 \\theta} = b^2\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u76f4\u7dda \\(\\ell : \\ mx+ny = 1\\) \u304c, \u6955\u5186 \\(C : \\ \\dfrac{x^2}{a^2} +\\dfrac{y^2}{b^2} =1 \\ ( a \\gt b \\gt 0 )\\) \u306b\u63a5\u3057\u306a\u304c\u3089\u52d5\u304f\u3068\u3059 … \u7d9a\u304d\u3092\u8aad\u3080
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n