\u5b9a\u6570 \\(a , b , c , d\\) \u306b\u5bfe\u3057\u3066, \u5e73\u9762\u4e0a\u306e\u70b9 \\((p,q)\\) \u3092\u70b9 \\((ap+bq , cp+dq)\\) \u306b\u79fb\u3059\u64cd\u4f5c\u3092\u8003\u3048\u308b.\r\n\u305f\u3060\u3057, \\((a,b,c,d) \\neq (1,0,0,1)\\) \u3067\u3042\u308b. \\(k\\) \u3092 \\(0\\) \u3067\u306a\u3044\u5b9a\u6570\u3068\u3059\u308b.\r\n\u653e\u7269\u7dda \\(C : \\ y = x^2-x+k\\) \u4e0a\u306e\u3059\u3079\u3066\u306e\u70b9\u306f, \u3053\u306e\u64cd\u4f5c\u306b\u3088\u3063\u3066 \\(C\\) \u4e0a\u306b\u79fb\u308b.<\/p>\r\n
(1)<\/strong>\u3000\\(a , b , c , d\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n (2)<\/strong>\u3000\\(C\\) \u4e0a\u306e\u70b9 A \u306b\u304a\u3051\u308b \\(C\\) \u306e\u63a5\u7dda\u3068, \u70b9 A \u3092\u3053\u306e\u64cd\u4f5c\u306b\u3088\u3063\u3066\u79fb\u3057\u305f\u70b9 A' \u306b\u304a\u3051\u308b \\(C\\) \u306e\u63a5\u7dda\u306f, \u539f\u70b9\u3067\u76f4\u4ea4\u3059\u308b. \u3053\u306e\u3068\u304d\u306e \\(k\\) \u306e\u5024\u304a\u3088\u3073\u70b9 A \u306e\u5ea7\u6a19\u3092\u3059\u3079\u3066\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \\(C\\) \u4e0a\u306e\u70b9\u3067\u3042\u308b\u70b9\u306b\u5bfe\u3057\u3066\u64cd\u4f5c\u3092\u8003\u3048\u308b. (2)<\/strong><\/p>\r\n (1)<\/strong> \u306e\u7d50\u679c\u3088\u308a, \u70b9 \\((p,q)\\) \u306f\u70b9 \\(( -p , 2p+q )\\) \u306b\u79fb\u308b.
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n
\r\n\u70b9 \\((0,k)\\) \u3092\u79fb\u3057\u305f\u70b9\u306f \\((bk , dk)\\) \u306a\u306e\u3067\r\n\\[\\begin{gather}\r\ndk = b^2k^2 -bk +k \\\\\r\n\\text{\u2234} \\quad b^2k^2 +(1-b-d)k = 0\r\n\\end{gather}\\]\r\n\u4efb\u610f\u306e \\(k\\) \u306b\u3064\u3044\u3066\u6210\u7acb\u3059\u308b\u306e\u3067\r\n\\[\\begin{align}\r\nb^2 = 0 & , \\ 1-b-d = 0 \\\\\r\n\\text{\u2234} \\quad b=0 & , \\ d=1\r\n\\end{align}\\]\r\n\u70b9 \\((k,k^2)\\) \u3092\u79fb\u3057\u305f\u70b9\u306f \\((ak , ck+k^2)\\) \u306a\u306e\u3067\r\n\\[\\begin{gather}\r\nck+k^2 = a^2k^2 -ak +k \\\\\r\n\\text{\u2234} \\quad \\left( a^2-1 \\right) k^2 +(1-a-c)k = 0\r\n\\end{gather}\\]\r\n\u3053\u308c\u304c, \u4efb\u610f\u306e \\(k\\) \u306b\u3064\u3044\u3066\u6210\u7acb\u3059\u308b\u306e\u3067\r\n\\[\r\na^2 -1 =0 , \\ 1-a-c=0\r\n\\]\r\n\\(a = 1\\) \u306e\u3068\u304d, \\(c=0\\) \u3068\u306a\u308b\u306e\u3067\u4e0d\u9069.
\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\r\na= -1 , \\ c=2\r\n\\]\r\n\u3088\u3063\u3066\r\n\\[\r\n(a,b,c,d) = \\underline{(-1 , 0 , 2 , 1)}\r\n\\]\r\n
\r\nA \\((t , t^2-t+k)\\) \u3068\u304a\u304f\u3068\r\n\\[\r\n\\text{A'} \\ ( -t , t^2+t+k )\r\n\\]\r\nA , A' \u306e\u63a5\u7dda\u3092\u305d\u308c\u305e\u308c \\(\\ell _ {\\text{A}} , \\ell _ {\\text{A'}}\\) \u3068\u304a\u304f\u3068\r\n\\[\r\ny' =2x-1\r\n\\]\r\n\u306a\u306e\u3067\r\n\\[\\begin{align}\r\n\\ell _ {\\text{A}} : \\ y & = (2t-1)(x-t) +t^2-t+k \\\\\r\n& = (2t-1)x -t^2+k \\\\\r\n\\ell _ {\\text{A'}} : \\ y & = (-2t-1)(x+t) +t^2+t+k \\\\\r\n& = (-2t-1)x -t^2+k\r\n\\end{align}\\]\r\n\u3053\u308c\u3089\u304c\u3068\u3082\u306b\u539f\u70b9\u3092\u901a\u308b\u306e\u3067\r\n\\[\r\n-t^2+k = 0 \\quad ... [1]\r\n\\]\r\n\u307e\u305f, \u76f4\u4ea4\u3059\u308b\u306e\u3067\r\n\\[\\begin{align}\r\n(2t-1)(-2t-1) & = -1 \\\\\r\n2t^2 -1 & = 0 \\\\\r\n\\text{\u2234} \\quad t = \\pm \\dfrac{\\sqrt{2}}{2} &\r\n\\end{align}\\]\r\n[1] \u306b\u4ee3\u5165\u3057\u3066\r\n\\[\r\nk = \\underline{\\dfrac{1}{2}}\r\n\\]\r\n\u3053\u306e\u3068\u304d, \\(y\\) \u5ea7\u6a19\u306f\r\n\\[\r\nt^2-t+k = 1 \\mp \\dfrac{\\sqrt{2}}{2}\r\n\\]\r\n\u3088\u3063\u3066, \u70b9 A \u306e\u5ea7\u6a19\u306f\r\n\\[\r\n\\underline{\\left( \\pm \\dfrac{\\sqrt{2}}{2} , 1 \\mp \\dfrac{\\sqrt{2}}{2} \\right) \\quad ( \\text{\u8907\u53f7\u540c\u9806} )}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u5b9a\u6570 \\(a , b , c , d\\) \u306b\u5bfe\u3057\u3066, \u5e73\u9762\u4e0a\u306e\u70b9 \\((p,q)\\) \u3092\u70b9 \\((ap+bq , cp+dq)\\) \u306b\u79fb\u3059\u64cd\u4f5c\u3092\u8003\u3048\u308b. \u305f\u3060\u3057, \\((a,b,c,d) \\neq (1,0,0,1)\\) … \u7d9a\u304d\u3092\u8aad\u3080