\\(xy\\) \u5e73\u9762\u4e0a\u306b\u6955\u5186 \\(C : \\ \\dfrac{x^2}{4} +y^2 = 1\\) \u304c\u3042\u308b.\r\n\u6b21\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n
(1)<\/strong>\u3000\u70b9 P \\(( a , b )\\) \u3092\u901a\u308b \\(C\\) \u306e\u63a5\u7dda\u304c \\(2\\) \u672c\u3042\u308a, \u305d\u308c\u3089\u304c\u76f4\u4ea4\u3059\u308b\u3068\u304d , \\(a , b\\) \u304c\u6e80\u305f\u3059\u6761\u4ef6\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n (2)<\/strong>\u3000\\(C\\) \u306b\u5916\u63a5\u3059\u308b\u9577\u65b9\u5f62\u306e\u3046\u3061, \\(x\\) \u5ea7\u6a19\u304c \\(1\\) \u3067 \\(y\\) \u5ea7\u6a19\u304c\u6b63\u3067\u3042\u308b\u9802\u70b9\u3092\u3082\u3064\u3082\u306e\u306e\u9762\u7a4d\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n 1*<\/strong>\u3000\\(a = \\pm 2\\) \u306e\u3068\u304d 2*<\/strong>\u3000\\(a \\neq \\pm 2\\) \u306e\u3068\u304d \u4ee5\u4e0a\u3088\u308a, \u6c42\u3081\u308b\u6761\u4ef6\u306f\r\n\\[\r\n\\underline{a^2 +b^2 = 5}\r\n\\]\r\n (2)<\/strong><\/p>\r\n (1)<\/strong> \u306e\u7d50\u679c\u3088\u308a, P \\(( 1 , 2 )\\) .
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n
\r\n
\r\n\u660e\u3089\u304b\u306b, \\(2\\) \u672c\u306e\u63a5\u7dda\u306f, \\(x = \\pm 2 , \\ y = \\pm 1\\) (\u8907\u53f7\u4efb\u610f) \u3067\u3042\u308a\r\n\\[\r\n( a , b ) = ( \\pm 2 , \\pm 1 ) \\quad ( \\ \\text{\u8907\u53f7\u4efb\u610f} \\ )\r\n\\]<\/li>\r\n
\r\nP \u306b\u304a\u3051\u308b\u63a5\u7dda\u306f\u50be\u304d \\(m\\) \u3092\u7528\u3044\u3066\r\n\\[\r\ny = m ( x-a ) +b\r\n\\]\r\n\u3068\u8868\u305b\u308b.
\r\n\\(C\\) \u306e\u5f0f\u306b\u4ee3\u5165\u3059\u308b\u3068\r\n\\[\\begin{align}\r\nx^2 +4 ( mx +b-ma )^2 & = 4 \\\\\r\n( 4m^2 +1 ) x^2 +8m ( b-ma ) x +4( b-ma )^2 -4 & = 0\r\n\\end{align}\\]\r\n\u3053\u306e\u65b9\u7a0b\u5f0f\u304c\u91cd\u89e3\u3092\u3082\u3064\u306e\u3067, \u5224\u5225\u5f0f\u3092 \\(D\\) \u3068\u304a\u3051\u3070\r\n\\[\\begin{align}\r\n\\dfrac{D}{4} = 16m^2 ( b-ma )^2 -4 ( 4m^2 +1 ) \\left\\{ ( b-ma )^2 -1 \\right\\} & = 0 \\\\\r\n-( b-ma )^2 +4m^2 +1 & = 0 \\\\\r\n\\text{\u2234} \\quad ( 4 -a^2 ) m^2 +2ab m +1 -b^2 & = 0 \\quad ... [1]\r\n\\end{align}\\]\r\n\\(4 -a^2 \\neq 0\\) \u306a\u306e\u3067, [1] \u306f \\(m\\) \u306b\u95a2\u3059\u308b \\(2\\) \u6b21\u65b9\u7a0b\u5f0f\u3067\u3042\u308a, \\(2\\) \u89e3\u3092 \\(s , t\\) \u3068\u304a\u3051\u3070, \u89e3\u3068\u4fc2\u6570\u306e\u95a2\u4fc2\u3088\u308a\r\n\\[\r\ns+t = -\\dfrac{2ab}{4 -a^2} , \\ st = \\dfrac{1 -b^2}{4 -a^2} \\quad ... [2]\r\n\\]\r\n\u63a5\u7dda\u304c\u76f4\u4ea4\u3059\u308b\u306e\u306f, \\(st = -1\\) \u306e\u3068\u304d\u306a\u306e\u3067\r\n\\[\\begin{align}\r\n\\dfrac{1 -b^2}{4 -a^2} & = -1 \\\\\r\n\\text{\u2234} \\quad a^2 +b^2 & = 5\r\n\\end{align}\\]\r\n\u3053\u308c\u306f, 1*<\/strong> \u306e\u3068\u304d\u3082\u307f\u305f\u3057\u3066\u3044\u308b.<\/p><\/li>\r\n<\/ol>\r\n
\r\nP \u3092\u901a\u308a\u76f4\u4ea4\u3059\u308b \\(2\\) \u672c\u306e\u63a5\u7dda\u306e\u5f0f\u306f\r\n\\[\r\ny = s (x-1) +2 , \\ y = t (x-1) +2\r\n\\]\r\n\u3068\u8868\u305b\u308b.
\r\n\u5404\u63a5\u7dda\u3068\u539f\u70b9\u3068\u306e\u8ddd\u96e2\u3092 \\(h_s , h_t\\) \u3068\u304a\u3051\u3070\r\n\\[\r\nh_s = \\dfrac{| 2-s |}{\\sqrt{s^2 +1}} , \\ h_t = \\dfrac{| 2-t |}{\\sqrt{t^2 +1}}\r\n\\]\r\n[2] \u306b \\(( a , b ) = ( 1 , 2 )\\) \u3092\u4ee3\u5165\u3059\u308c\u3070\r\n\\[\\begin{align}\r\ns+t & = -\\dfrac{2 \\cdot 2}{4-1^2} = -\\dfrac{4}{3} \\ , \\\\\r\nst & = \\dfrac{1-2^2}{4-1^2} = -1\r\n\\end{align}\\]\r\n\u3055\u3089\u306b\r\n\\[\\begin{align}\r\ns^2 t^2 & = 1 \\ , \\\\\r\ns^2 +t^2 & = (s+t)^2 -2st \\\\\r\n& = \\dfrac{16}{9} +2 = \\dfrac{34}{9}\r\n\\end{align}\\]\r\n\u3053\u308c\u3089\u3092\u7528\u3044\u308c\u3070, \u6c42\u3081\u308b\u9762\u7a4d \\(S\\) \u306f\r\n\\[\\begin{align}\r\nS & = 4 h_s h_t \\\\\r\n& = 4 \\cdot \\dfrac{| (2-s) (2-t) |}{\\sqrt{( s^2 +1 ) ( t^2 +1 )}} \\\\\r\n& = 4 \\cdot \\dfrac{\\left| 4 -2 \\left( -\\dfrac{4}{3} \\right) -1 \\right|}{1 +\\dfrac{34}{9} +1} \\\\\r\n& = 4 \\cdot \\dfrac{\\dfrac{17}{3}}{\\dfrac{2 \\sqrt{13}}{3}} \\\\\r\n& = \\underline{\\dfrac{34}{\\sqrt{13}}}\r\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\\(xy\\) \u5e73\u9762\u4e0a\u306b\u6955\u5186 \\(C : \\ \\dfrac{x^2}{4} +y^2 = 1\\) \u304c\u3042\u308b. \u6b21\u306e\u554f\u3044\u306b\u7b54\u3048\u3088. (1)\u3000\u70b9 P \\(( a , b )\\) \u3092\u901a\u308b \\(C\\) \u306e\u63a5\u7dda\u304c \\(2\\) \u672c\u3042\u308a, … \u7d9a\u304d\u3092\u8aad\u3080