\\(xy\\) \u5e73\u9762\u4e0a\u306b \\(3\\) \u3064\u306e\u66f2\u7dda\r\n\\[\\begin{align}\r\nC _ 1 & : \\ y = x^2 \\\\\r\nC _ 2 & : \\ y = 2(x-1)^2 +3 \\\\\r\nC _ 3 & : \\ y = -(x-a)^2+b \\quad ( \\ a , b \\text{\u306f\u5b9f\u6570} )\r\n\\end{align}\\]\r\n\u304c\u3042\u308b. \\(C _ 2\\) \u3068 \\(C _ 3\\) \u306f\u305f\u3060 \\(1\\) \u3064\u306e\u70b9\u3092\u5171\u6709\u3057\u3066\u3044\u308b. \u6b21\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n
(1)<\/strong>\u3000\\(C _ 1 , C _ 2\\) \u306f\u5171\u6709\u70b9\u3092\u3082\u305f\u306a\u3044\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n (2)<\/strong>\u3000\\(b\\) \u3092 \\(a\\) \u306e\u5f0f\u3067\u8868\u305b.<\/p><\/li>\r\n (3)<\/strong>\u3000\\(C _ 1\\) \u3068 \\(C _ 3\\) \u3067\u56f2\u307e\u308c\u308b\u90e8\u5206\u306e\u9762\u7a4d\u3092 \\(S(a)\\) \u3068\u3059\u308b. \\(S(a)\\) \u3092\u6700\u5c0f\u306b\u3059\u308b \\(a\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \\(C _ 1 , C _ 2\\) \u306e\u5f0f\u3088\u308a \\(y\\) \u3092\u6d88\u53bb\u3059\u308b\u3068\r\n\\[\\begin{gather}\r\nx^2 =2(x-1)^2 +3 \\\\\r\n\\text{\u2234} \\quad x^2 -4x+5 & = 0 \\quad ... [1]\r\n\\end{gather}\\]\r\n[1] \u306e\u5224\u5225\u5f0f\u3092 \\(D _ 1\\) \u3068\u304a\u3051\u3070\r\n\\[\r\n\\dfrac{D _ 1}{4} = 2^2 -1 \\cdot 5 = -1 \\lt 0\r\n\\]\r\n\u3088\u3063\u3066, [1]\u306f\u5b9f\u6570\u89e3\u3092\u3082\u305f\u305a, \u984c\u610f\u306f\u793a\u3055\u308c\u305f.<\/p>\r\n (2)<\/strong><\/p>\r\n \\(C _ 2 , C _ 3\\) \u306e\u5f0f\u3088\u308a \\(y\\) \u3092\u6d88\u53bb\u3059\u308b\u3068\r\n\\[\\begin{gather}\r\n2(x-1)^2 +3= -(x-a)^2+b \\\\\r\n\\text{\u2234} \\quad 3x^2 -2(a+2)x +a^2-b+5 = 0 \\quad ... [2]\r\n\\end{gather}\\]\r\n[2] \u304c\u91cd\u89e3\u3092\u3082\u3064\u306e\u3067, \u5224\u5225\u5f0f\u3092 \\(D _ 2\\) \u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\n\\dfrac{D _ 2}{4} = (a+2)^2 -3(a^2-b+5) & = 0 \\\\\r\n-2a^2 +4a -11 +3b & = 0 \\\\\r\n\\text{\u2234} \\quad = \\underline{\\dfrac{2a^2-4a+11}{3}} &\r\n\\end{align}\\]\r\n (3)<\/strong><\/p>\r\n \\(C _ 1 , C _ 3\\) \u306e\u5f0f\u3088\u308a \\(y\\) \u3092\u6d88\u53bb\u3059\u308b\u3068\r\n\\[\\begin{gather}\r\nx^2 = -(x-a)^2+b \\\\\r\n\\text{\u2234} \\quad 2x^2 -2ax +a^2 -b = 0\\quad ... [3]\r\n\\end{gather}\\]\r\n[3] \u306e\u5224\u5225\u5f0f\u3092 \\(D _ 3\\) \u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\n\\dfrac{D _ 3}{4} & = a^2 -2(a^2-b) =-a^2 +2b \\\\\r\n& = \\dfrac{1}{3} \\left( a^2-8a+22 \\right) \\\\\r\n& = \\dfrac{1}{3} \\left\\{ (a-4)^2 +6 \\right\\} \\gt 0\r\n\\end{align}\\]\r\n\u306a\u306e\u3067, [3] \u306f\u7570\u306a\u308b \\(2\\) \u3064\u306e\u5b9f\u6570\u89e3\u3092\u3082\u3061, \\(C _ 1\\) \u3068 \\(C _ 3\\) \u306f\u7570\u306a\u308b \\(2\\) \u3064\u306e\u4ea4\u70b9\u3092\u3082\u3064.
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n
\r\n\u3053\u308c\u3089\u306e \\(x\\) \u5ea7\u6a19\u3092 \\(\\alpha , \\beta \\ ( \\alpha \\lt \\beta )\\) \u3068\u304a\u304f\u3068, \u89e3\u3068\u4fc2\u6570\u306e\u95a2\u4fc2\u3088\u308a\r\n\\[\r\n\\alpha +\\beta = 2a , \\ \\alpha \\beta = a^2-b\r\n\\]\r\n\u3053\u308c\u3089\u3092\u7528\u3044\u308c\u3070\r\n\\[\\begin{align}\r\nS(a) & = \\displaystyle\\int _ {\\alpha}^{\\beta} \\left\\{ x^2 +(x-a)^2 -b \\right\\} \\, dx \\\\\r\n& = \\dfrac{1}{3} \\left| \\beta -\\alpha \\right|^3 \\\\\r\n& = \\dfrac{1}{3} \\left| (2a)^2 -4 \\left( a^2-b \\right) \\right|^{\\frac{3}{2}} \\\\\r\n& = \\dfrac{1}{27} \\left| 2a^2-4a+11 \\right|^{\\frac{3}{2}} \\\\\r\n& = \\dfrac{1}{27} \\left\\{ 2(a-1)^2+9 \\right\\}^{\\frac{3}{2}}\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, \\(S(a)\\) \u3092\u6700\u5c0f\u306b\u3059\u308b \\(a\\) \u306e\u5024\u306f\r\n\\[\r\na = \\underline{2}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(xy\\) \u5e73\u9762\u4e0a\u306b \\(3\\) \u3064\u306e\u66f2\u7dda \\[\\begin{align} C _ 1 & : \\ y = x^2 \\\\ C _ 2 & : \\ y = 2(x-1)^2 +3 \\\\ C _ 3 & : \\ y = - … \u7d9a\u304d\u3092\u8aad\u3080