\u5ea7\u6a19\u5e73\u9762\u4e0a\u306e\u653e\u7269\u7dda \\(C\\) \u3092 \\(y =x^2+1\\) \u3067\u5b9a\u3081\u308b.\r\n\\(s , t\\) \u306f\u5b9f\u6570\u3068\u3057, \\(t \\lt 0\\) \u3092\u6e80\u305f\u3059\u3068\u3059\u308b. \u70b9 \\((s,t)\\) \u304b\u3089\u653e\u7269\u7dda \\(C\\) \u3078\u5f15\u3044\u305f\u63a5\u7dda\u3092 \\(l _ 1 , l _ 2\\) \u3068\u3059\u308b.<\/p>\r\n
(1)<\/strong>\u3000\\(l _ 1 , l _ 2\\) \u306e\u65b9\u7a0b\u5f0f\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n (2)<\/strong>\u3000\\(a\\) \u3092\u6b63\u306e\u5b9f\u6570\u3068\u3059\u308b. \u653e\u7269\u7dda \\(C\\) \u3068\u76f4\u7dda \\(l _ 1 , l _ 2\\) \u3067\u56f2\u307e\u308c\u308b\u9818\u57df\u306e\u9762\u7a4d\u304c \\(a\\) \u3068\u306a\u308b \\((s,t)\\) \u3092\u5168\u3066\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \\(C\\) \u306e\u5f0f\u3088\u308a, \\(y' = 2x\\) \u306a\u306e\u3067, \u63a5\u70b9\u306e \\(x\\) \u5ea7\u6a19\u304c \\(p\\) \u3067\u3042\u308b\u63a5\u7dda \\(l\\) \u306e\u5f0f\u306f\r\n\\[\\begin{align}\r\ny & = 2p (x-p) +p^2+1 \\\\\r\n& = 2px -p^2+1\r\n\\end{align}\\]\r\n\u3053\u308c\u304c \\((s,t)\\) \u3092\u901a\u308b\u306e\u3067\r\n\\[\\begin{align}\r\nt & = 2ps -p^2+1 \\\\\r\n\\text{\u2234} \\quad p^2 & -2sp +t-1 = 0 \\quad ... [1]\r\n\\end{align}\\]\r\n[1] \u306e\u5224\u5225\u5f0f \\(D\\) \u306b\u3064\u3044\u3066\r\n\\[\r\n\\dfrac{D}{4} = s^2 -t+1 \\gt 0 \\quad ( \\ \\text{\u2235} \\ t \\lt 0 \\ )\r\n\\]\r\n\u306a\u306e\u3067, [1] \u306f \\(2\\) \u3064\u306e\u7570\u306a\u308b\u89e3\u3092\u3082\u3064. (2)<\/strong><\/p>\r\n \\(\\alpha = s +\\sqrt{s^2 -t+1}\\) , \\(\\beta = s -\\sqrt{s^2 -t+1}\\) \u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\nl _ 1 & : \\ y = 2 \\alpha x -\\alpha^2 +1 , \\\\\r\nl _ 2 & : \\ y = 2 \\beta x -\\beta^2 +1\r\n\\end{align}\\]\r\n\u307e\u305f\r\n\\[\r\ns = \\dfrac{\\alpha +\\beta}{2}\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \u653e\u7269\u7dda \\(C\\) \u3068\u76f4\u7dda \\(l _ 1 , l _ 2\\) \u306b\u56f2\u307e\u308c\u308b\u9818\u57df\u306e\u9762\u7a4d\u3092 \\(S\\) \u3068\u304a\u3051\u3070\r\n\\[\\begin{align}\r\nS & = \\displaystyle\\int _ {\\beta}^{s} \\left\\{ x^2+1 -( 2 \\beta x -\\beta^2 +1 ) \\right\\} \\, dx \\\\\r\n& \\qquad + \\displaystyle\\int _ {s}^{\\alpha} \\left\\{ x^2+1 -( 2 \\alpha x -\\alpha^2 +1 ) \\right\\} \\, dx \\\\\r\n& = \\displaystyle\\int _ {\\beta}^{\\frac{\\alpha +\\beta}{2}} ( x -\\beta )^2 \\, dx +\\displaystyle\\int _ {\\frac{\\alpha +\\beta}{2}}^{\\alpha} ( x -\\alpha )^2 \\, dx \\\\\r\n& = \\left[ \\dfrac{( x -\\beta )^3}{3} \\right] _ {\\beta}^{\\frac{\\alpha +\\beta}{2}} +\\left[ \\dfrac{( x -\\alpha )^3}{3} \\right] _ {\\frac{\\alpha +\\beta}{2}}^{\\alpha} \\\\\r\n& = \\dfrac{( \\alpha -\\beta )^3}{12} \\\\\r\n& = \\dfrac{2 \\left( s^2 -t+1 \\right)^{\\frac{3}{2}}}{3}\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(S = a\\) \u3068\u306a\u308b\u306e\u306f\r\n\\[\\begin{gather}\r\n\\dfrac{2 \\left( s^2 -t+1 \\right)^{\\frac{3}{2}}}{3} = a \\\\\r\ns^2 -t+1 = \\left( \\dfrac{3a}{2} \\right)^{\\frac{2}{3}} \\\\\r\n\\text{\u2234} \\quad t = s^2+1 -\\left( \\dfrac{3a}{2} \\right)^{\\frac{2}{3}} \\quad ... [1]\r\n\\end{gather}\\]\r\n\\(t \\lt 0\\) , \\(s^2 \\geqq 0\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\n1 -\\left( \\dfrac{3a}{2} \\right)^{\\frac{2}{3}} & \\lt 0 \\\\\r\n\\text{\u2234} \\quad a \\gt \\dfrac{2}{3} &\r\n\\end{align}\\]\r\n\u3053\u306e\u3068\u304d, [1]\u306f \\(t \\lt 0\\) \u306a\u308b\u89e3\u3092\u3082\u3064.
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n
\r\n[1] \u3092\u89e3\u304f\u3068\r\n\\[\r\np = s \\pm \\sqrt{s^2 -t+1}\r\n\\]\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u65b9\u7a0b\u5f0f\u306f\r\n\\[\\begin{align}\r\nl _ 1 & : \\ \\underline{y = 2 \\left( s +\\sqrt{s^2 -t+1} \\right) x -\\left( s +\\sqrt{s^2 -t+1} \\right)^2 +1} , \\\\\r\nl _ 2 & : \\ \\underline{y = 2 \\left( s -\\sqrt{s^2 -t+1} \\right) x -\\left( s -\\sqrt{s^2 -t+1} \\right)^2 +1}\r\n\\end{align}\\]\r\n
\r\n\u4e00\u65b9, \\(0 \\lt a \\leqq \\dfrac{2}{3}\\) \u306e\u3068\u304d\u306f\r\n\\[\r\nt \\geqq 0\r\n\\]\r\n\u306a\u306e\u3067, [1] \u306f \\(t \\lt 0\\) \u306a\u308b\u89e3\u3092\u3082\u305f\u306a\u3044.
\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u89e3\u306f\r\n\\[\r\n\\underline{\\left\\{ \\begin{array}{ll} \\text{\u89e3\u306a\u3057} & \\left( \\ 0 \\lt a \\leqq \\dfrac{2}{3} \\text{\u306e\u3068\u304d} \\right) \\\\ t = s^2+1 -\\left( \\dfrac{3a}{2} \\right)^{\\frac{2}{3}} \\lt 0 & \\left( \\ a \\gt \\dfrac{2}{3} \\text{\u306e\u3068\u304d} \\right) \\end{array} \\right.}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u5ea7\u6a19\u5e73\u9762\u4e0a\u306e\u653e\u7269\u7dda \\(C\\) \u3092 \\(y =x^2+1\\) \u3067\u5b9a\u3081\u308b. \\(s , t\\) \u306f\u5b9f\u6570\u3068\u3057, \\(t \\lt 0\\) \u3092\u6e80\u305f\u3059\u3068\u3059\u308b. \u70b9 \\((s,t)\\) \u304b\u3089\u653e\u7269\u7dda \\(C\\) \u3078\u5f15\u3044\u305f\u63a5\u7dda\u3092 \\(l … \u7d9a\u304d\u3092\u8aad\u3080