\u95a2\u6570 \\(y = x(x-1)(x-3)\\) \u306e\u30b0\u30e9\u30d5\u3092 \\(C\\) , \u539f\u70b9 O \u3092\u901a\u308b\u50be\u304d \\(t\\) \u306e\u76f4\u7dda\u3092 \\(\\ell\\) \u3068\u3057, \\(C\\) \u3068 \\(\\ell\\) \u304c O \u4ee5\u5916\u306b\u5171\u6709\u70b9\u3092\u3082\u3064\u3068\u3059\u308b.\r\n\\(C\\) \u3068 \\(\\ell\\) \u306e\u5171\u6709\u70b9\u3092 O , P , Q \u3068\u3057, \\(\\overrightarrow{\\text{OP}}\\) \u3068 \\(\\overrightarrow{\\text{OQ}}\\) \u306e\u7a4d\u3092 \\(g(t)\\) \u3068\u304a\u304f.\r\n\u305f\u3060\u3057, \u305d\u308c\u3089\u5171\u6709\u70b9\u306e \\(1\\) \u3064\u304c\u63a5\u70b9\u3067\u3042\u308b\u5834\u5408\u306f, O , P , Q \u306e\u3046\u3061 \\(2\\) \u3064\u304c\u4e00\u81f4\u3057\u3066, \u305d\u306e\u63a5\u70b9\u3067\u3042\u308b\u3068\u3059\u308b.\r\n\u95a2\u6570 \\(g(t)\\) \u306e\u5897\u6e1b\u3092\u8abf\u3079, \u305d\u306e\u6975\u5024\u3092\u6c42\u3081\u3088.<\/p>\r\n
\u307e\u305a, \\(C\\) \u3068 \\(\\ell\\) \u304c O \u4ee5\u5916\u306b\u5171\u6709\u70b9\u3092\u3082\u3064\u6761\u4ef6\u3092\u8003\u3048\u308b.
\r\n\\(\\ell : \\ y = tx\\) \u3068 \\(C\\) \u306e\u5f0f\u304b\u3089 \\(y\\) \u3092\u6d88\u53bb\u3059\u308b\u3068\r\n\\[\\begin{align}\r\nx(x-1)(x-3) & = tx \\\\\r\n\\text{\u2234} \\quad x ( x^2-4x-t+3 ) & = 0\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\r\nx^2-4x-t+3 = 0 \\quad ... [1]\r\n\\]\r\n\u304c \\(0\\) \u4ee5\u5916\u306e\u89e3\u3092\u6301\u3066\u3070\u3088\u3044.
\r\n[1] \u306b \\(x=0\\) \u3092\u4ee3\u5165\u3059\u308b\u3068\r\n\\[\\begin{align}\r\n-t +3 & = 0 \\\\\r\n\\text{\u2234} \\quad t & = 3\r\n\\end{align}\\]\r\n\u3053\u306e\u3068\u304d, [1] \u306b\u4ee3\u5165\u3057\u3066\r\n\\[\\begin{align}\r\nx^2 -4x & = 0 \\\\\r\nx (x-4) &= 0 \\\\\r\n\\text{\u2234} \\quad x & = 0 , 4\r\n\\end{align}\\]\r\n\u306a\u306e\u3067, \\(x=0\\) \u4ee5\u5916\u306b\u89e3\u3092\u3082\u3064.
\r\n\\(t \\neq 3\\) \u306e\u3068\u304d, [1] \u306e\u5224\u5225\u5f0f \\(D\\) \u306b\u3064\u3044\u3066\r\n\\[\\begin{align}\r\n\\dfrac{D}{4} = 2^2 & -(-t+3) \\geqq 0 \\\\\r\n\\text{\u2234} \\quad t & \\geqq -1\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \u4ee5\u4e0b\u3067\u306f \\(t \\geqq -1\\) \u306e\u3082\u3068\u3067\u8003\u3048\u308b.
\r\nP , Q \u306e \\(x\\) \u5ea7\u6a19\u3092 \\(p , q\\) \u3068\u304a\u3051\u3070, [1] \u3068\u89e3\u3068\u4fc2\u6570\u306e\u95a2\u4fc2\u304b\u3089\r\n\\[\r\np+q = 4 , \\ pq = -t+3 \\quad ... [2]\r\n\\]\r\n[2] \u3092\u7528\u3044\u308b\u3068\r\n\\[\\begin{align}\r\ng(t) & = |p| \\sqrt{1+t^2} \\cdot |q| \\sqrt{1+t^2} \\\\\r\n& = | pq | \\left( 1+t^2 \\right) \\\\\r\n& = | t-3 | \\left( 1+t^2 \\right) \\\\\r\n& = \\left\\{ \\begin{array}{ll} (t-3) \\left( 1+t^2 \\right) & ( t \\geqq 3 \\text{\u306e\u3068\u304d} ) \\\\ -(t-3) \\left( 1+t^2 \\right) & ( -1 \\leqq t \\lt 3 \\text{\u306e\u3068\u304d} ) \\end{array} \\right.\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\\begin{align}\r\ng'(t) & = \\pm \\left\\{ \\left( 1+t^2 \\right) +2t (t-3) \\right\\} \\\\\r\n& = \\pm \\left( 3t^2 -6t +1 \\right)\r\n\\end{align}\\]\r\n\\(g'(t) = 0\\) \u3092\u3068\u304f\u3068\r\n\\[\r\nt = \\dfrac{3 \\pm \\sqrt{3^2 -3}}{3} = 1 \\pm \\dfrac{\\sqrt{6}}{3}\r\n\\]\r\n\\(-1 \\lt 1 -\\dfrac{\\sqrt{6}}{3} \\lt 1 +\\dfrac{\\sqrt{6}}{3} \\lt 3\\) \u3067\u3042\u308a\r\n\\[\\begin{align}\r\n\\left| g \\left( 1 \\pm \\dfrac{\\sqrt{6}}{3} \\right) \\right| & = \\left| \\left( -2 \\pm \\dfrac{\\sqrt{6}}{3} \\right) \\left( \\dfrac{8}{3} \\pm \\dfrac{2 \\sqrt{6}}{3} \\right) \\right|\\\\\r\n& = \\left| -\\dfrac{16}{3} +\\dfrac{4}{3} \\pm \\left( \\dfrac{8}{9} -\\dfrac{4}{3} \\right) \\right| \\\\\r\n& = \\left| -4 \\mp \\dfrac{\\sqrt{6}}{3} \\right| , \\\\\r\ng(3) & = 0 , \\ g( -1 ) = 4 \\cdot 2 = 8\r\n\\end{align}\\]\r\n\u4ee5\u4e0a\u3088\u308a, \\(g(t)\\) \u306e\u5897\u6e1b\u306f\u4e0b\u8868\u306e\u3088\u3046\u306b\u306a\u308b.\r\n\\[\r\n\\begin{array}{c|cccccccc} t & -1 & \\cdots & 1 -\\dfrac{\\sqrt{6}}{3} & \\cdots & 1 +\\dfrac{\\sqrt{6}}{3} & \\cdots & 3 & \\cdots \\\\ \\hline g'(t) & & - & 0 & + & 0 & - & & + \\\\ \\hline g(t) & 8 & \\searrow & 4 -\\dfrac{\\sqrt{6}}{3} & \\nearrow & 4 +\\dfrac{\\sqrt{6}}{3} & \\searrow & 0 & \\nearrow \\end{array}\r\n\\]\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u6975\u5024\u306f\r\n\\[\r\n\\underline{g \\left( 1 -\\dfrac{\\sqrt{6}}{3} \\right) = 4 -\\dfrac{\\sqrt{6}}{3} , \\ g \\left( 1 +\\dfrac{\\sqrt{6}}{3} \\right) = 4 +\\dfrac{\\sqrt{6}}{3} , \\ g(3) = 0}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u95a2\u6570 \\(y = x(x-1)(x-3)\\) \u306e\u30b0\u30e9\u30d5\u3092 \\(C\\) , \u539f\u70b9 O \u3092\u901a\u308b\u50be\u304d \\(t\\) \u306e\u76f4\u7dda\u3092 \\(\\ell\\) \u3068\u3057, \\(C\\) \u3068 \\(\\ell\\) \u304c O \u4ee5\u5916\u306b\u5171\u6709\u70b9\u3092\u3082\u3064\u3068\u3059\u308b. \\(C … \u7d9a\u304d\u3092\u8aad\u3080