\u653e\u7269\u7dda \\(C : \\ y = x^2\\) \u4e0a\u306e\u7570\u306a\u308b \\(2\\) \u70b9 P \\(( t , t^2 )\\) , Q \\(( s , s^2 ) \\quad ( s \\lt t )\\) \u306b\u304a\u3051\u308b\u63a5\u7dda\u306e\u4ea4\u70b9\u3092 R \\(( X , Y )\\) \u3068\u3059\u308b.<\/p>\r\n
(1)<\/strong>\u3000\\(X , Y\\) \u3092 \\(t , s\\) \u3092\u7528\u3044\u3066\u793a\u305b.<\/p><\/li>\r\n (2)<\/strong>\u3000\u70b9 P, Q \u304c \\(\\angle \\text{PRQ} =\\dfrac{\\pi}{4}\\) \u3092\u6e80\u305f\u3057\u306a\u304c\u3089 \\(C\\) \u4e0a\u3092\u52d5\u304f\u3068\u304d, \u70b9 R \u306f\u53cc\u66f2\u7dda\u4e0a\u3092\u52d5\u304f\u3053\u3068\u3092\u793a\u3057, \u304b\u3064, \u305d\u306e\u53cc\u66f2\u7dda\u306e\u65b9\u7a0b\u5f0f\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \\(y = x^2\\) \u3088\u308a, \\(y' =2x\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\n\\text{PR} : \\ y = 2t(x-t) +t^2 = 2tx -t^2 , \\\\\r\n\\text{QR} : \\ y = 2sx -s^2\r\n\\end{align}\\]\r\nPR , QR \u306e\u5f0f\u3088\u308a\r\n\\[\\begin{align}\r\n2tx -t^2 & = 2sx -s^2 \\\\\r\n2(t-s)x & = t^2 -s^2 \\\\\r\n\\text{\u2234} \\quad x & = \\dfrac{t+s}{2} \\quad ( \\ \\text{\u2235} \\ t \\neq s \\ )\r\n\\end{align}\\]\r\n\u307e\u305f\r\n\\[\r\ny = 2t \\cdot \\dfrac{t+s}{2} -t^2 = st\r\n\\]\r\n\u3088\u3063\u3066\r\n\\[\r\n( X , Y ) = \\underline{\\left( \\dfrac{t+s}{2} , st \\right)}\r\n\\]\r\n (2)<\/strong><\/p>\r\n PR , QR \u306e \\(x\\) \u8ef8\u6b63\u65b9\u5411\u3068\u306a\u3059\u89d2\u3092 \\(\\alpha , \\beta\\) \u3068\u304a\u304f\u3068\r\n\\[\r\n\\tan \\alpha = 2t , \\ \\tan \\beta = 2s\r\n\\]\r\n\u307e\u305f \\(s \\lt t\\) \u306a\u306e\u3067, \\(\\beta \\lt \\alpha\\) .
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n
\r\n\\(\\angle \\text{PRQ} = \\dfrac{\\pi}{4}\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\n\\tan (\\beta -\\alpha) & = \\tan \\dfrac{\\pi}{4} \\\\\r\n\\dfrac{\\tan \\beta -\\tan \\alpha}{1 +\\tan \\alpha \\tan \\beta} & = 1 \\\\\r\n\\text{\u2234} \\quad 2(s-t) & = 1+4st \\quad ... [1]\r\n\\end{align}\\]\r\n\u3053\u3053\u3067 \\(s \\lt t\\) \u3088\u308a, \\(1+4st \\lt 0\\) ... [2] .
\r\n[1] \u306e\u4e21\u8fba\u3092\u5e73\u65b9\u3059\u308b\u3068\r\n\\[\r\n4(t+s)^2 -16st = (1+4st)^2\r\n\\]\r\n(1)<\/strong> \u306e\u7d50\u679c\u3092\u4ee3\u5165\u3059\u308c\u3070\r\n\\[\\begin{align}\r\n16X^2 -16Y = 1+8Y & +16Y^2 \\\\\r\n16X^2 -16Y^2 -24Y & = 1 \\\\\r\n16X^2 -16 \\left( Y +\\dfrac{3}{4}\\right)^2 & = -8 \\\\\r\n\\text{\u2234} \\quad \\left( Y +\\dfrac{3}{4}\\right)^2 -X^2 & = \\dfrac{1}{2}\r\n\\end{align}\\]\r\n\u307e\u305f [2] \u3088\u308a, \\(Y \\lt -\\dfrac{1}{4}\\) .
\r\n\u3088\u3063\u3066, R \u306f\r\n\\[\r\n\\text{\u53cc\u66f2\u7dda} : \\ \\underline{\\left( y +\\dfrac{3}{4}\\right)^2 -x^2= \\dfrac{1}{2}}\r\n\\]\r\n\u306e \\(y \\lt -\\dfrac{1}{4}\\) \u306e\u90e8\u5206, \u3064\u307e\u308a\u4e0b\u534a\u5206\u3092\u52d5\u304f.<\/p>\r\n","protected":false},"excerpt":{"rendered":"\u653e\u7269\u7dda \\(C : \\ y = x^2\\) \u4e0a\u306e\u7570\u306a\u308b \\(2\\) \u70b9 P \\(( t , t^2 )\\) , Q \\(( s , s^2 ) \\quad ( s \\lt t )\\) \u306b\u304a\u3051\u308b\u63a5\u7dda\u306e\u4ea4\u70b9\u3092 R \\(( X … \u7d9a\u304d\u3092\u8aad\u3080