(1)<\/strong>\u3000\u70b9 P \u304b\u3089\u66f2\u7dda \\(C\\) \u306b\u63a5\u7dda\u304c\u3061\u3087\u3046\u3069 \\(3\\) \u672c\u5f15\u3051\u308b\u3088\u3046\u306a\u70b9 \\((a,b)\\) \u306e\u5b58\u5728\u3059\u308b\u9818\u57df\u3092\u56f3\u793a\u305b\u3088.<\/p><\/li>\r\n (2)<\/strong>\u3000\u70b9 P \u304b\u3089\u66f2\u7dda \\(C\\) \u306b\u63a5\u7dda\u304c\u3061\u3087\u3046\u3069 \\(2\\) \u672c\u5f15\u3051\u308b\u3068\u3059\u308b. \\(2\\) \u3064\u306e\u63a5\u70b9\u3092 A , B \u3068\u3057\u305f\u3068\u304d, \\(\\angle \\text{APB}\\) \u304c \\(90^{\\circ}\\) \u3088\u308a\u5c0f\u3055\u304f\u306a\u308b\u305f\u3081\u306e \\(a\\) \u3068 \\(b\\) \u306e\u6761\u4ef6\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \\(C\\) \u306e\u5f0f\u3088\u308a\r\n\\[\r\ny' = 3x^2-a^2\r\n\\]\r\n\u306a\u306e\u3067, \\(C\\) \u4e0a\u306e \\(x\\) \u5ea7\u6a19\u304c \\(t\\) \u3067\u3042\u308b\u70b9\u306b\u304a\u3051\u308b\u63a5\u7dda\u306e\u5f0f\u306f\r\n\\[\\begin{align}\r\ny & = ( 3t^2-a^2 )( x-t ) +t^3 -a^2 t +a^3 \\\\\r\n& = ( 3t^2-a^2 ) x -2t^3 +a^3\r\n\\end{align}\\]\r\n\u3053\u308c\u304c\u70b9 P \u3092\u901a\u308b\u3068\u304d\r\n\\[\\begin{align}\r\n( 3t^2-a^2 ) b -2t^3 +a^3 & = 0 \\\\\r\n\\text{\u2234} \\quad 2t^3 -3bt^2 +a^2(b-a) & = 0 \\quad ... [1]\r\n\\end{align}\\]\r\n\\(t\\) \u306e\u65b9\u7a0b\u5f0f [1] \u304c \\(3\\) \u3064\u306e\u7570\u306a\u308b\u89e3\u3092\u3082\u3064\u6761\u4ef6\u3092\u6c42\u3081\u308c\u3070\u3088\u3044. (2)<\/strong><\/p>\r\n \u63a5\u7dda\u304c\u3061\u3087\u3046\u3069 \\(2\\) \u672c\u5f15\u3051\u308b\u306e\u306f, (1)<\/strong> \u306e\u7d4c\u904e\u3088\u308a\r\n\\[\r\nf(0) = 0 \\ \\text{\u307e\u305f\u306f} \\ f(b) = 0\r\n\\]\r\n\u3068\u306a\u308b\u3068\u304d\u3067\u3042\u308b.
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n
\r\n[1] \u306e\u5de6\u8fba\u3092 \\(f(t)\\) \u3068\u304a\u304f\u3068\r\n\\[\r\nf'(t) = 6t^2-6bt = 6t(t-b)\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(f(t)\\) \u306e\u5897\u6e1b\u306f\u4e0b\u8868\u306e\u3088\u3046\u306b\u306a\u308b.\r\n\\[\r\n\\begin{array}{c|ccccc} t & \\cdots & 0 & \\cdots & b & \\cdots \\\\ \\hline f'(t) & + & 0 & - & 0 & + \\\\ \\hline f(t) & \\nearrow & \\text{\u6975\u5927} & \\searrow & \\text{\u6975\u5c0f} & \\nearrow \\end{array}\r\n\\]\r\n\u6c42\u3081\u308b\u6761\u4ef6\u306f\r\n\\[\r\nf(0) \\gt 0 \\quad ... [2] \\ \\text{\u304b\u3064} \\ f(b) \\lt 0 \\quad ... [3]\r\n\\]\r\n[2] \u3088\u308a\r\n\\[\\begin{align}\r\nf(0) & = a^2(b-a) \\gt 0 \\\\\r\n\\text{\u2234} \\quad b & \\gt a \\quad ... [4]\r\n\\end{align}\\]\r\n[4] \u3067\u3042\u308c\u3070, [3] \u306b\u3064\u3044\u3066\r\n\\[\\begin{align}\r\nf(b) & = a^2(b-a) -b^3 \\\\\r\n& \\lt a^2b -b^3 \\\\\r\n& = b(a^2-b^2) \\lt 0 \\quad ... [5]\r\n\\end{align}\\]\r\n\u3067, [3] \u3082\u6210\u7acb\u3059\u308b.
\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u9818\u57df\u306f\u4e0b\u56f3\u659c\u7dda\u90e8\uff08\u5883\u754c\u3068\u25cb\u306f\u542b\u307e\u306a\u3044\uff09.<\/p>\r\n![\"\"](\"\/\/www.roundown.net\/nyushi\/wp-content\/uploads\/tohoku_r_2010_02_01.png\" "\\"tohoku_r_2010_02_01\\"")
\r\n
\r\n\u3057\u304b\u3057, [5] \u3088\u308a \\(b \\gt a\\) \u306e\u3068\u304d, \\(f(b) \\lt 0\\) .
\r\n\u307e\u305f, \\(b \\leqq a\\) \u306e\u3068\u304d\r\n\\[\r\nf(b) = -a^2(a-b) -b^3 \\leqq -b^3 \\lt 0\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066\u5e38\u306b\r\n\\[\r\nf(b) \\lt 0\r\n\\]\r\n\u3086\u3048\u306b, \\(f(0) = 0\\) \u3068\u306a\u308b\u3068\u304d\u306b\u3064\u3044\u3066\u306e\u307f\u8003\u3048\u308c\u3070\u3088\u3044.\r\n\\[\\begin{align}\r\nf(0) = a^2(b-a) & = 0 \\\\\r\n\\text{\u2234} \\quad b & = a\r\n\\end{align}\\]\r\n\u3053\u306e\u3068\u304d [1] \u306e\u89e3\u306f\r\n\\[\\begin{align}\r\n2t^3 -3at^2 & = 0 \\\\\r\nt^2 ( 2t-3a ) & = 0 \\\\\r\n\\text{\u2234} \\quad t & = 0 , \\dfrac{3a}{2}\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\r\n\\text{A} \\ \\left( 0, a^3 \\right) , \\quad B \\ \\left( \\dfrac{3a}{2} , \\dfrac{23a^3}{8} \\right) , \\quad P \\ ( a, 0 )\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(\\angle \\text{APB}\\) \u304c \\(90^{\\circ}\\) \u3088\u308a\u5c0f\u3055\u304f\u306a\u308b\u6761\u4ef6\u306f, \u30d9\u30af\u30c8\u30eb \\(\\overrightarrow{\\text{PA}}\\) \u3068 \\(\\overrightarrow{\\text{PB}}\\) \u306e\u5185\u7a4d\u3092\u8003\u3048\u3066\r\n\\[\\begin{align}\r\n\\overrightarrow{\\text{PA}} \\cdot \\overrightarrow{\\text{PB}} & \\gt 0 \\\\\r\n(-a) \\cdot \\dfrac{a}{2} + a^3 \\cdot \\dfrac{23a^3}{8} & \\gt 0 \\\\\r\na^4 & \\gt \\dfrac{4}{23} \\\\\r\n\\text{\u2234} \\quad a & \\gt \\sqrt[4]{\\dfrac{4}{23}}\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u6761\u4ef6\u306f\r\n\\[\r\n\\underline{a = b \\gt \\sqrt[4]{\\dfrac{4}{23}}}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(a , b\\) \u3092\u6b63\u306e\u5b9f\u6570\u3068\u3059\u308b. \u66f2\u7dda \\(C : \\ y = x^3-a^2x+a^3\\) \u3068\u70b9 P \\((b , 0)\\) \u3092\u8003\u3048\u308b. \u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088. (1)\u3000\u70b9 P \u304b\u3089\u66f2\u7dda \\(C\\) \u306b\u63a5\u7dda\u304c\u3061\u3087\u3046 … \u7d9a\u304d\u3092\u8aad\u3080