\\(k \\gt 0\\) \u3068\u3059\u308b.\r\n\u5186 \\(C\\) \u3092 \\(x^2 +(y-1)^2 = 1\\) \u3068\u3057, \u653e\u7269\u7dda \\(S\\) \u3092 \\(y = \\dfrac{1}{k} x^2\\) \u3068\u3059\u308b.<\/p>\r\n
(1)<\/strong>\u3000\\(C\\) \u3068 \\(S\\) \u304c\u5171\u6709\u70b9\u3092\u3061\u3087\u3046\u3069 \\(3\\) \u500b\u6301\u3064\u3068\u304d\u306e \\(k\\) \u306e\u7bc4\u56f2\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n (2)<\/strong>\u3000\\(k\\) \u304c (1)<\/strong> \u306e\u7bc4\u56f2\u3092\u52d5\u304f\u3068\u304d, \\(C\\) \u3068 \\(S\\) \u306e\u5171\u6709\u70b9\u306e\u3046\u3061\u3067 \\(x\\) \u5ea7\u6a19\u304c\u6b63\u306e\u70b9\u3092 P \u3068\u3059\u308b.\r\nP \u306b\u304a\u3051\u308b \\(S\\) \u306e\u63a5\u7dda\u3068 \\(S\\) \u3068 \\(y\\) \u8ef8\u3068\u306b\u3088\u3063\u3066\u56f2\u307e\u308c\u308b\u9818\u57df\u306e\u9762\u7a4d\u306e\u6700\u5927\u5024\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \\(C\\) \u3068 \\(S\\) \u306e\u5f0f\u304b\u3089 \\(y\\) \u3092\u6d88\u53bb\u3057\u3066\r\n\\[\\begin{align}\r\nx^2 +\\left( \\dfrac{x^2}{k} -1 \\right)^2 & = 0 \\\\\r\n\\dfrac{x^4}{k^2} +\\left( -\\dfrac{2}{k} +1 \\right) x^2 & = 0 \\\\\r\n\\text{\u2234} \\quad x^2 \\left\\{ x^2 -k (2-k) \\right\\} & = 0\r\n\\end{align}\\]\r\n\u3053\u308c\u304c \\(3\\) \u3064\u306e\u7570\u306a\u308b\u5b9f\u6570\u89e3\u3092\u3082\u3064\u6761\u4ef6\u3092\u6c42\u3081\u308c\u3070\u3088\u3044\u306e\u3067\r\n\\[\\begin{gather}\r\nk (2-k) \\gt 0 \\\\\r\n\\text{\u2234} \\quad \\underline{0 \\lt k \\lt 2}\r\n\\end{gather}\\]\r\n (2)<\/strong><\/p>\r\n P \u306e \\(x\\) \u5ea7\u6a19\u3092 \\(t\\) \u3068\u304a\u3051\u3070\r\n\\[\r\nt = \\sqrt{k (2-k)}\r\n\\]\r\n\u3053\u306e\u70b9\u306b\u304a\u3051\u308b \\(S\\) \u306e\u63a5\u7dda\u306e\u5f0f\u306f \\(y' = \\dfrac{2x}{k}\\) \u3088\u308a\r\n\\[\\begin{align}\r\ny & = \\dfrac{2t}{k} (x-t) +\\dfrac{t^2}{k} \\\\\r\n& = \\dfrac{2t}{k} x -\\dfrac{t^2}{k}\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\\begin{align}\r\nS & = \\displaystyle\\int _ {0}^{t} \\left( \\dfrac{x^2}{k} -\\dfrac{2t}{k} x +\\dfrac{t^2}{k} \\right) \\, dx \\\\\r\n& = \\dfrac{1}{k} \\displaystyle\\int _ {0}^{t} (x-t)^2 \\, dx \\\\\r\n& = \\dfrac{1}{k} \\left[ \\dfrac{1}{3} (x-t)^3 \\right] _ {0}^{t} \\\\\r\n& = \\dfrac{t^3}{3k} \\\\\r\n& = \\dfrac{\\sqrt{k^3 (2-k)^3}}{3k} \\\\\r\n& = \\dfrac{1}{3} \\sqrt{\\underline{k (2-k)^3} _ {[1]}}\r\n\\end{align}\\]\r\n[1] \u304c\u6700\u5927\u3068\u306a\u308b\u3068\u304d, \\(S\\) \u3082\u6700\u5927\u3068\u306a\u308b.
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n
\r\n\\(s = 2-k\\) \u3068\u304a\u304f\u3068 (1)<\/strong> \u306e\u7d50\u679c\u3088\u308a, \\(0 \\lt s \\lt 2\\) .
\r\n\u307e\u305f [1] \u306b\u3064\u3044\u3066\r\n\\[\r\n[1] = (2-s) s^3 = -s^4 +2s^3\r\n\\]\r\n\u3053\u308c\u3092 \\(f(s)\\) \u3068\u304a\u304f\u3068\r\n\\[\r\nf'(s) = -4s^3 +6s^2 = -2s^2 (2s-3)\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(f(s)\\) \u306e\u5897\u6e1b\u306f\u4e0b\u8868\u306e\u901a\u308a.\r\n\\[\r\n\\begin{array}{c|ccccc} s & (0) & \\cdots & \\dfrac{3}{2} & \\cdots & (2) \\\\ \\hline f'(s) & & + & 0 & - & & \\\\ \\hline f(s) & & \\nearrow & \\text{\u6700\u5927} & \\searrow & \\end{array}\r\n\\]\r\n\u3053\u3053\u3067\r\n\\[\r\nf \\left( \\dfrac{3}{2} \\right) = \\dfrac{1}{2} \\left( \\dfrac{3}{2} \\right)^3 = \\dfrac{27}{16}\r\n\\]\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u6700\u5927\u5024\u306f\r\n\\[\r\n\\dfrac{1}{3} \\sqrt{f \\left( \\dfrac{3}{2} \\right)} = \\dfrac{1}{3} \\cdot \\dfrac{3 \\sqrt{3}}{4} = \\underline{\\dfrac{\\sqrt{3}}{4}}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(k \\gt 0\\) \u3068\u3059\u308b. \u5186 \\(C\\) \u3092 \\(x^2 +(y-1)^2 = 1\\) \u3068\u3057, \u653e\u7269\u7dda \\(S\\) \u3092 \\(y = \\dfrac{1}{k} x^2\\) \u3068\u3059\u308b. (1)\u3000\\(C\\) \u3068 \\(S\\ … \u7d9a\u304d\u3092\u8aad\u3080