\\(xy\\) \u5e73\u9762\u4e0a\u306b, \\(b \\lt a^2\\) \u3092\u307f\u305f\u3059\u70b9 A \\((a,b)\\) \u304c\u3042\u308b.\r\n\u66f2\u7dda \\(C : \\ y = x^2\\) \u4e0a\u306b\u70b9 P \u3092\u3068\u308a, \u7dda\u5206 AP \u3092 \\(k : k-1 \\ ( k \\gt 1 )\\) \u306b\u5916\u5206\u3059\u308b\u70b9\u3092 Q \u3068\u3059\u308b.\r\nP \u304c \\(C\\) \u4e0a\u3092\u52d5\u304f\u3068\u304d\u306b\u3067\u304d\u308b Q \u306e\u8ecc\u8de1\u3092 \\(C'\\) \u3068\u3059\u308b. \u6b21\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n
(1)<\/strong>\u3000\\(C'\\) \u306e\u65b9\u7a0b\u5f0f\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n (2)<\/strong>\u3000\\(C\\) \u3068 \\(C'\\) \u306f \\(2\\) \u3064\u306e\u70b9\u3067\u4ea4\u308f\u308b\u3053\u3068\u3092\u793a\u3057, \\(C\\) \u3068 \\(C'\\) \u3067\u56f2\u307e\u308c\u308b\u90e8\u5206\u306e\u9762\u7a4d\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n P \\(( t , t^2 )\\) , Q \\(( X , Y )\\) \u3068\u304a\u304f. (2)<\/strong><\/p>\r\n \\(C , C'\\) \u306e\u5f0f\u304b\u3089 \\(y\\) \u3092\u6d88\u53bb\u3059\u308b\u3068\r\n\\[\\begin{gather}\r\nkx^2 = x^2 +2(k-1)ax +(k-1)^2a^2 -k(k-1)b \\\\\r\n\\text{\u2234} \\quad x^2 -2ax -(k-1)a^2 +kb = 0 \\quad ... [3]\r\n\\end{gather}\\]\r\n[3] \u306e\u5224\u5225\u5f0f\u3092 \\(D\\) \u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\n\\dfrac{D}{4} & = a^2 +(k-1)a^2 -kb\\ \\\\\r\n& = k ( a^2-b ) \\gt 0 \\quad ( \\ \\text{\u2235} \\ k>1 , \\ b \\lt a^2 )\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(C\\) \u3068 \\(C'\\) \u306f \\(2\\) \u3064\u306e\u70b9\u3067\u4ea4\u308f\u308b.
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n
\r\n\u6761\u4ef6\u3088\u308a, \\(\\overrightarrow{\\text{AQ}} = k \\overrightarrow{\\text{AP}}\\) \u306a\u306e\u3067\r\n\\[\\begin{gather}\r\n\\left( \\begin{array}{c} X-a \\\\ Y-b \\end{array} \\right) = k \\left( \\begin{array}{c} t-a \\\\ t^2-b \\end{array} \\right) \\\\\r\n\\text{\u2234} \\quad \\left\\{ \\begin{array}{ll} X = k(t-a)+a & ... [1] \\\\ Y = k(t^2-b)+b & ... [2] \\end{array} \\right.\r\n\\end{gather}\\]\r\n[1] \u3088\u308a\r\n\\[\r\nkt = X +(k-1) a\r\n\\]\r\n\u3053\u308c\u3092 [2] \u306b\u4ee3\u5165\u3057\u3066\r\n\\[\\begin{align}\r\nkY & = \\left\\{ X +(k-1)a \\right\\}^2 -k^2b +kb \\\\\r\n\\text{\u2234} \\quad Y & = \\dfrac{1}{k} \\left\\{ X +(k-1)a \\right\\}^2 -(k-1)b\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, \\(C'\\) \u306e\u65b9\u7a0b\u5f0f\u306f\r\n\\[\r\n\\underline{y = \\dfrac{1}{k} \\left\\{ x +(k-1)a \\right\\}^2 -(k-1)b}\r\n\\]\r\n
\r\n[3] \u306e \\(2\\) \u3064\u306e\u89e3\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 = -(k-1)a^2 +kb\r\n\\]\r\n\u3053\u308c\u3092\u7528\u3044\u308c\u3070, \u6c42\u3081\u308b\u9762\u7a4d \\(S\\) \u306f\r\n\\[\\begin{align}\r\nS & = \\displaystyle\\int _ {\\alpha}^{\\beta} \\left[ x^2 -\\dfrac{1}{k} \\left\\{ x +(k-1)a \\right\\}^2 +(k-1)b \\right] \\, dx \\\\\r\n& = -\\dfrac{k-1}{k} \\displaystyle\\int _ {\\alpha}^{\\beta} ( x-\\alpha )( x-\\beta ) \\, dx \\\\\r\n& = \\dfrac{k-1}{6k} ( \\beta -\\alpha )^3 \\\\\r\n& = \\dfrac{k-1}{6k} \\left\\{ ( \\alpha +\\beta )^2 -4 \\alpha \\beta \\right\\}^{\\frac{3}{2}} \\\\\r\n& = \\dfrac{k-1}{6k} \\left\\{ 4a^2 +4(k-1)a^2 -4kb \\right\\}^{\\frac{3}{2}} \\\\\r\n& = \\underline{\\dfrac{4 \\sqrt{k} (k-1) (a^2-b)^{\\frac{3}{2}}}{3}}\r\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\\(xy\\) \u5e73\u9762\u4e0a\u306b, \\(b \\lt a^2\\) \u3092\u307f\u305f\u3059\u70b9 A \\((a,b)\\) \u304c\u3042\u308b. \u66f2\u7dda \\(C : \\ y = x^2\\) \u4e0a\u306b\u70b9 P \u3092\u3068\u308a, \u7dda\u5206 AP \u3092 \\(k : k-1 \\ ( k \\gt … \u7d9a\u304d\u3092\u8aad\u3080