\\(a\\) \u3092\u6b63\u306e\u5b9f\u6570\u3068\u3059\u308b.\r\n\u653e\u7269\u7dda \\(y = x^2\\) \u3092 \\(C_1\\) , \u653e\u7269\u7dda \\(y = -x^2 +4ax -4 a^2 +4 a^4\\) \u3092 \\(C_2\\) \u3068\u3059\u308b.\r\n\u4ee5\u4e0b\u306e\u554f\u306b\u7b54\u3048\u3088.<\/p>\r\n
(1)<\/strong>\u3000\u70b9 \\(( t , t^2 )\\) \u306b\u304a\u3051\u308b \\(C_1\\) \u306e\u63a5\u7dda\u306e\u65b9\u7a0b\u5f0f\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n (2)<\/strong>\u3000\\(C_1\\) \u3068 \\(C_2\\) \u304c\u7570\u306a\u308b \\(2\\) \u3064\u306e\u5171\u901a\u63a5\u7dda \\(\\ell , \\ell '\\) \u3092\u6301\u3064\u3088\u3046\u306a \\(a\\) \u306e\u7bc4\u56f2\u3092\u6c42\u3081\u3088.\r\n\u305f\u3060\u3057, \\(C_1\\) \u3068 \\(C_2\\) \u306e\u5171\u901a\u63a5\u7dda\u3068\u306f, \\(C_1\\) \u3068 \\(C_2\\) \u306e\u4e21\u65b9\u306b\u63a5\u3059\u308b\u76f4\u7dda\u306e\u3053\u3068\u3067\u3042\u308b.<\/p><\/li>\r\n<\/ol>\r\n \u4ee5\u4e0b, \\(a\\) \u306f (2)<\/strong> \u3067\u6c42\u3081\u305f\u7bc4\u56f2\u306b\u3042\u308b\u3068\u3057,\r\n\\(\\ell , \\ell '\\) \u3092 \\(C_1\\) \u3068 \\(C_2\\) \u306e\u7570\u306a\u308b \\(2\\) \u3064\u306e\u5171\u901a\u63a5\u7dda\u3068\u3059\u308b.<\/p>\r\n (3)<\/strong>\u3000\\(\\ell , \\ell '\\) \u306e\u4ea4\u70b9\u306e\u5ea7\u6a19\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n (4)<\/strong>\u3000\\(C_1\\) \u3068 \\(\\ell , \\ell '\\) \u3067\u56f2\u307e\u308c\u305f\u9818\u57df\u3092 \\(D_1\\) \u3068\u3057, \u4e0d\u7b49\u5f0f \\(x \\leqq a\\) \u306e\u8868\u3059\u9818\u57df\u3092 \\(D_2\\) \u3068\u3059\u308b.\r\n\\(D_1\\) \u3068 \\(D_2\\) \u306e\u5171\u901a\u90e8\u5206\u306e\u9762\u7a4d \\(S(a)\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n (5)<\/strong>\u3000\\(S(a)\\) \u3092 (4)<\/strong> \u306e\u901a\u308a\u3068\u3059\u308b. \\(a\\) \u304c (2)<\/strong> \u3067\u6c42\u3081\u305f\u7bc4\u56f2\u3092\u52d5\u304f\u3068\u304d, \\(S(a)\\) \u306e\u6700\u5927\u5024\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \\(C_1\\) \u306e\u5f0f\u3088\u308a, \\(y' = 2x\\) \u306a\u306e\u3067, \u63a5\u7dda\u306e\u5f0f\u306f\r\n\\[\\begin{align}\r\ny & = 2t (x-t) + t^2 \\\\\r\n& = 2tx -t^2\r\n\\end{align}\\]\r\n\u3059\u306a\u308f\u3061\r\n\\[\r\n\\underline{y = 2tx -t^2}\r\n\\]\r\n (2)<\/strong><\/p>\r\n (1)<\/strong> \u3067\u6c42\u3081\u305f\u5f0f\u3068 \\(C_2\\) \u306e\u5f0f\u304b\u3089 \\(y\\) \u3092\u6d88\u53bb\u3057\u3066\r\n\\[\\begin{gather}\r\n2tx -t^2 = -x^2 +4ax -4a^2 +4a^4 \\\\\r\n\\text{\u2234} \\quad x^2 +2( t -2a ) x -t^2 -4a^2 +4a^4 = 0\r\n\\end{gather}\\]\r\n\u3053\u308c\u304c\u91cd\u89e3\u3092\u3082\u3064\u306e\u3067, \u5224\u5225\u5f0f \\(D_1\\) \u306b\u3064\u3044\u3066\r\n\\[\\begin{align}\r\n\\dfrac{D_1}{4} & = ( t -2a )^2 +t^2 +4a^2 -4a^4 \\\\\r\n& = 2t^2 -4at +4a^4 = 0\\\\\r\n\\text{\u2234} \\quad & t^2 -2a t +2a^4 = 0 \\quad ... [1]\r\n\\end{align}\\]\r\n\u3053\u308c\u304c\u7570\u306a\u308b \\(2\\) \u5b9f\u6570\u89e3\u3092\u3082\u3064\u306e\u3067, \u5224\u5225\u5f0f \\(D_2\\) \u306b\u3064\u3044\u3066\r\n\\[\\begin{align}\r\n\\dfrac{D_2}{4} = a^2 -2a^4 & \\gt 0 \\\\\r\n1 -2a^2 & \\gt 0 \\quad ( \\ \\text{\u2235} \\ a^2 \\gt 0 \\ ) \\\\\r\n\\text{\u2234} \\quad & \\underline{0 \\lt a \\lt \\dfrac{1}{\\sqrt{2}}}\r\n\\end{align}\\]\r\n (3)<\/strong><\/p>\r\n[1] \u306e \\(2\\) \u89e3\u3092 \\(p , q \\ (p\\lt q )\\) \u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\n\\ell \\ : \\ y & = 2px -p^2 \\\\\r\n\\ell' \\ : \\ y & = 2qx -q^2\r\n\\end{align}\\]\r\n\u3053\u308c\u3092\u3068\u304f\u3068\r\n\\[\\begin{align}\r\n2px -p^2 & = 2qx -q^2 \\\\\r\n2( p-q ) x & = p^2 -q^2 \\\\\r\n\\text{\u2234} \\quad x & = \\dfrac{p+q}{2} \\quad ( \\ \\text{\u2235} \\ p \\neq q \\ )\r\n\\end{align}\\]\r\n\u3086\u3048\u306b\r\n\\[\r\ny = 2p \\cdot \\dfrac{p+q}{2} -p^2 = pq\r\n\\]\r\n\u89e3\u3068\u4fc2\u6570\u306e\u95a2\u4fc2\u304b\u3089, [1] \u3088\u308a\r\n\\[\r\np+q = 2a \\ , \\ pq = 2a^4 \\quad ... [2]\r\n\\]\r\n\u3088\u3063\u3066, \u4ea4\u70b9\u306e\u5ea7\u6a19\u306f\r\n\\[\r\n\\underline{( a , 2a^4 )}\r\n\\]\r\n (4)<\/strong><\/p>\r\n[2] \u3088\u308a\r\n\\[\\begin{align}\r\n(q-p)^2 & = (p+q)^2 -4pq \\\\\r\n& = 4a^2 -8a^2 = 4 a^2 ( 1 -2a^2 )\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\\begin{align}\r\nS(a) & = \\displaystyle\\int _ {p}^{\\frac{p+q}{2}} ( x^2 -2p x +p^2 ) \\, dx \\\\\r\n& = \\left[ \\dfrac{(x-p)^3}{3} \\right] _ {p}^{\\frac{p+q}{2}} = \\dfrac{(q-p)^3}{24} \\\\\r\n& = \\dfrac{8 a^3 ( 1 -2a^2 )^{\\frac{3}{2}}}{24} \\\\\r\n& = \\underline{\\dfrac{1}{3} a^3 ( 1 -2a^2 )^{\\frac{3}{2}}} \\\\\r\n\\end{align}\\]\r\n (5)<\/strong><\/p>\r\n \\(u = a^2\\) \u3068\u304a\u304f\u3068, \\(0 \\lt u \\lt \\dfrac{1}{2}\\) .\r\n
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n
\r\n\\(f(u) = u ( 1 -2u )\\) \u3068\u304a\u3051\u3070\r\n\\[\r\nf(u) = -2 \\left( u -\\dfrac{1}{4} \\right)^2 +\\dfrac{1}{8}\r\n\\]\r\n\u3088\u3063\u3066, \\(u = \\dfrac{1}{4}\\) \u3059\u306a\u308f\u3061 \\(a = \\dfrac{1}{2}\\) \u306e\u3068\u304d, \\(S(a)\\) \u306f\u6700\u5927\u3068\u306a\u308a, \u305d\u306e\u5024\u306f\r\n\\[\r\n\\dfrac{1}{3} \\left\\{ f \\left( \\dfrac{1}{2} \\right) \\right\\}^{\\frac{3}{2}} = \\dfrac{1}{3} \\left( \\dfrac{1}{8} \\right)^{\\frac{3}{2}} = \\underline{\\dfrac{\\sqrt{2}}{96}}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(a\\) \u3092\u6b63\u306e\u5b9f\u6570\u3068\u3059\u308b. \u653e\u7269\u7dda \\(y = x^2\\) \u3092 \\(C_1\\) , \u653e\u7269\u7dda \\(y = -x^2 +4ax -4 a^2 +4 a^4\\) \u3092 \\(C_2\\) \u3068\u3059\u308b. \u4ee5\u4e0b\u306e\u554f\u306b\u7b54\u3048\u3088. (1)\u3000\u70b9 … \u7d9a\u304d\u3092\u8aad\u3080