\u539f\u70b9 O \\((0,0)\\) \u3092\u4e2d\u5fc3\u3068\u3059\u308b\u534a\u5f84 \\(1\\) \u306e\u5186\u306b, \u5186\u5916\u306e\u70b9 P \\(( x _ 0 , y _ 0 )\\) \u304b\u3089 \\(2\\) \u672c\u306e\u63a5\u7dda\u3092\u5f15\u304f.<\/p>\r\n
(1)<\/strong>\u3000\\(2\\) \u3064\u306e\u63a5\u70b9\u306e\u4e2d\u70b9\u3092 Q \u3068\u3059\u308b\u3068\u304d, \u70b9 Q \u306e\u5ea7\u6a19 \\(( x _ 1 , y _ 1 )\\) \u3092\u70b9 P \u306e\u5ea7\u6a19 \\(( x _ 0 , y _ 0 )\\) \u3092\u7528\u3044\u3066\u8868\u305b. \u307e\u305f \\(\\text{OP} \\cdot \\text{OQ} = 1\\) \u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n (2)<\/strong>\u3000\u70b9 P \u304c\u76f4\u7dda \\(x+y = 2\\) \u4e0a\u3092\u52d5\u304f\u3068\u304d, \u70b9 Q \u306e\u8ecc\u8de1\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \\(2\\) \u3064\u306e\u63a5\u70b9\u3092 A \\((a,b)\\) , B \\((c,d)\\) \u3068\u304a\u304f\u3068\r\n\\[\r\n\\text{Q} \\ \\left( \\dfrac{a+c}{2} , \\dfrac{b+d}{2} \\right) \\ .\r\n\\]\r\n\u307e\u305f\r\n\\[\r\n\\text{PA} : \\ ax+by = 1 , \\ \\text{PB} : \\ cx+dy = 1 \\ .\r\n\\]\r\n\u3068\u3082\u306b P \\(( x _ 0 , y _ 0 )\\) \u3092\u901a\u308b\u306e\u3067\r\n\\[\r\na x _ 0 +b y _ 0 = 1 , \\ c x _ 0 +d y _ 0 = 1 \\ .\r\n\\]\r\n\u3053\u308c\u306f\u76f4\u7dda \\(x _ 0 x +y _ 0 y = 1\\) \u304c \\(2\\) \u70b9 A , B \u3092\u901a\u308b\u3053\u3068\u3092\u8868\u3059. (2)<\/strong><\/p>\r\n (1)<\/strong> \u306e\u7d50\u679c\u3088\u308a\r\n\\[\\begin{align}\r\nx _ 1 = \\dfrac{x _ 0}{{x _ 0}^2 +{y _ 0}^2} & , \\ y _ 1 = \\dfrac{y _ 0}{{x _ 0}^2 +{y _ 0}^2} , \\\\\r\n{x _ 1}^2 +{y _ 1}^2 & = \\dfrac{1}{{x _ 0}^2 +{y _ 0}^2} \\quad ... [4] \\ .\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\r\nx _ 0 = \\dfrac{x _ 1}{{x _ 1}^2 +{y _ 1}^2} , \\ y _ 0 = \\dfrac{y _ 1}{{x _ 1}^2 +{y _ 1}^2} \\quad ... [5] \\ .\r\n\\]\r\n\u305f\u3060\u3057, [4] \u3088\u308a, \\({x _ 1}^2 +{y _ 1}^2 \\neq 0\\) \u3059\u306a\u308f\u3061 \\(( x _ 1 , y _ 1 ) \\neq (0,0)\\) .
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n
\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\r\n\\text{AB} : \\ x _ 0 x +y _ 0 y = 1 \\quad ... [1] \\ .\r\n\\]\r\n\u3053\u308c\u3068, \u5186 \\(x^2+y^2 = 1 \\quad ... [2]\\) \u306e\u4ea4\u70b9\u304c\u70b9 A , B \u3067\u3042\u308b\u306e\u3067, [1] [2] \u3088\u308a, \\(y\\) \u3092\u6d88\u53bb\u3059\u308b\u3068\r\n\\[\\begin{align}\r\n{y _ 0}^2 x^2 +( 1 -x _ 0 x )^2 & = {y _ 0}^2 \\\\\r\n\\text{\u2234} \\quad ( {x _ 0}^2 +{y _ 0}^2 ) x^2 -2 x _ 0 x +1 -{y _ 0}^2 & = 0 \\ .\r\n\\end{align}\\]\r\n\u3053\u306e\u65b9\u7a0b\u5f0f\u306e \\(2\\) \u3064\u306e\u89e3\u304c \\(a , c\\) \u306a\u306e\u3067, \u89e3\u3068\u4fc2\u6570\u306e\u95a2\u4fc2\u3088\u308a\r\n\\[\r\na+c = \\dfrac{2 x _ 0}{{x _ 0}^2 +{y _ 0}^2} \\ .\r\n\\]\r\n\u3086\u3048\u306b\r\n\\[\r\nx _ 1 = \\dfrac{a+c}{2} = \\dfrac{x _ 0}{{x _ 0}^2 +{y _ 0}^2} \\ .\r\n\\]\r\n\u540c\u69d8\u306b, [1] [2] \u304b\u3089 \\(y\\) \u3092\u6d88\u53bb\u3059\u308c\u3070,\r\n\\[\\begin{align}\r\n( 1 -y _ 0 y )^2 +{x _ 0}^2 y^2 & = {x _ 0}^2 \\\\\r\n\\text{\u2234} \\quad ( {x _ 0}^2 +{y _ 0}^2 ) x^2 -2 y _ 0 y +1 -{x _ 0}^2 & = 0 \\ .\r\n\\end{align}\\]\r\n\u3053\u306e\u65b9\u7a0b\u5f0f\u306e \\(2\\) \u3064\u306e\u89e3\u304c \\(b , d\\) \u306a\u306e\u3067, \u89e3\u3068\u4fc2\u6570\u306e\u95a2\u4fc2\u3088\u308a\r\n\\[\r\nb+d = \\dfrac{2 y _ 0}{{x _ 0}^2 +{y _ 0}^2} \\ .\r\n\\]\r\n\u3086\u3048\u306b\r\n\\[\r\ny _ 1 = \\dfrac{b+d}{2} = \\dfrac{y _ 0}{{x _ 0}^2 +{y _ 0}^2} \\ .\r\n\\]\r\n\u3088\u3063\u3066, Q \u306e\u5ea7\u6a19\u306f\r\n\\[\r\n\\underline{\\left( \\dfrac{x _ 0}{{x _ 0}^2 +{y _ 0}^2} , \\dfrac{y _ 0}{{x _ 0}^2 +{y _ 0}^2} \\right)} \\ .\r\n\\]\r\n\u307e\u305f\r\n\\[\r\n\\overrightarrow{\\text{OQ}} = \\dfrac{1}{\\left| \\overrightarrow{\\text{OP}} \\right|^2} \\overrightarrow{\\text{OP}} \\quad ... [3] \\ .\r\n\\]\r\n\u306a\u306e\u3067\r\n\\[\\begin{align}\r\n\\left| \\overrightarrow{\\text{OQ}} \\right| & = \\dfrac{1}{\\left| \\overrightarrow{\\text{OP}} \\right|} \\\\\r\n\\text{\u2234} \\quad \\left| \\overrightarrow{\\text{OP}} \\right| \\left| \\overrightarrow{\\text{OQ}} \\right| & = 1 \\ .\r\n\\end{align}\\]\r\n
\r\n[5] \u3092 \\(x _ 0 +y _ 0 = 2\\) \u306b\u4ee3\u5165\u3059\u308c\u3070\r\n\\[\\begin{align}\r\nx _ 1 + y _ 1 = 2 \\left( {x _ 1}^2 +{y _ 1}^2 \\right) & \\\\\r\n{x _ 1}^2 -\\dfrac{1}{2} x _ 1 +{y _ 1}^2 -\\dfrac{1}{2} y _ 1 = 0 \\\\\r\n\\text{\u2234} \\quad \\left( x _ 1 -\\dfrac{1}{4} \\right)^2 +\\left( y _ 1 -\\dfrac{1}{4} \\right)^2 & = \\dfrac{1}{8} \\ .\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, Q \u306e\u8ecc\u8de1\u306f\r\n\\[\r\n\\underline{\\text{\u5186} : \\ \\left( x -\\dfrac{1}{4} \\right)^2 +\\left( y -\\dfrac{1}{4} \\right)^2 = \\dfrac{1}{8} \\quad ( \\text{\u305f\u3060\u3057, \u539f\u70b9\u306f\u9664\u304f} )} \\ .\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u539f\u70b9 O \\((0,0)\\) \u3092\u4e2d\u5fc3\u3068\u3059\u308b\u534a\u5f84 \\(1\\) \u306e\u5186\u306b, \u5186\u5916\u306e\u70b9 P \\(( x _ 0 , y _ 0 )\\) \u304b\u3089 \\(2\\) \u672c\u306e\u63a5\u7dda\u3092\u5f15\u304f. (1)\u3000\\(2\\) \u3064\u306e\u63a5\u70b9\u306e\u4e2d\u70b9\u3092 Q \u3068\u3059\u308b\u3068\u304d, … \u7d9a\u304d\u3092\u8aad\u3080