{"id":484,"date":"2012-11-24T14:43:23","date_gmt":"2012-11-24T05:43:23","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=484"},"modified":"2021-09-23T21:13:02","modified_gmt":"2021-09-23T12:13:02","slug":"thr201102","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/thr201102\/","title":{"rendered":"\u6771\u5317\u5927\u7406\u7cfb2011\uff1a\u7b2c2\u554f"},"content":{"rendered":"
\n

\\(a\\) \u3092\u5b9f\u6570\u3068\u3059\u308b. \u5186 \\(C\\) \u306f\u70b9 \\((a, -a)\\) \u3067\u76f4\u7dda \\(y = -x\\) \u3092\u63a5\u7dda\u306b\u3082\u3061, \u70b9 \\((0,1)\\) \u3092\u901a\u308b\u3082\u306e\u3068\u3059\u308b. \\(C\\) \u306e\u4e2d\u5fc3\u3092 P \\((X,Y)\\) \u3068\u3057\u3066, \u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(X , Y\\) \u3092 \\(a\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(a\\) \u304c\u52d5\u304f\u3068\u304d\u306e\u70b9P\u306e\u8ecc\u8de1\u3068\u76f4\u7dda \\(y = 1\\) \u3067\u56f2\u307e\u308c\u308b\u56f3\u5f62\u306e\u9762\u7a4d\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

    \u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n\"\"\r\n

    (1)<\/strong><\/p>\r\n

    \u5186 \\(C\\) \u306f, \u70b9 P\u3068\u70b9 \\((1,0)\\) \u3092\u901a\u308b\u306e\u3067\r\n\\[\\begin{align}\r\n(X-a)^2 +(Y+a)^2 & = X^2 +(Y-1)^2 \\\\\r\n-2aX +a^2 +2aY +a^2 & = -2Y +1 \\\\\r\n\\text{\u2234} \\quad 2aX -2(a+1)Y -2a^2 +1 & = 0 \\quad ... [1] \\ .\r\n\\end{align}\\]\r\n\u70b9 P \u306f\u63a5\u70b9 \\((a, -a)\\) \u3092\u901a\u308b\u50be\u304d \\(1\\) \u306e\u76f4\u7dda\u4e0a\u306b\u3042\u308b\u306e\u3067\r\n\\[\r\nY = X-2a \\quad ... [2] \\ .\r\n\\]\r\n[1] \u306b [2] \u3092\u4ee3\u5165\u3057\u3066\r\n\\[\\begin{align}\r\n2aX -2(a+1)(X-2a) -2a^2 +1 & =0 \\\\\r\n-2X +4a^2 +4a -2a^2 +1 & = 0 \\\\\r\n\\text{\u2234} \\quad X = \\underline{a^2 +2a +\\dfrac{1}{2}} & \\ .\r\n\\end{align}\\]\r\n[2] \u3088\u308a\r\n\\[\r\nY = \\underline{a^2 +\\dfrac{1}{2}} \\ .\r\n\\]\r\n

    (2)<\/strong><\/p>\r\n

    (1)<\/strong> \u306e\u7d50\u679c\u3088\u308a\r\n\\[\\begin{align}\r\nY & = \\left( \\dfrac{X-Y}{2} \\right)^2 +\\dfrac{1}{2} \\\\\r\n(X-Y)^2 & = 2(Y-2) \\ .\r\n\\end{align}\\]\r\n\u4e21\u8fba\u306e\u5e73\u65b9\u6839\u3092\u3068\u3063\u3066\r\n\\[\\begin{align}\r\nX-Y & = \\pm \\sqrt{2(2Y-1)} \\\\\r\n\\text{\u2234} \\quad X & = Y \\pm \\sqrt{2(2Y-1)} \\ .\r\n\\end{align}\\]\r\n\\(Y = 1\\) \u3068\u304a\u304f\u3068\r\n\\[\r\na = \\pm \\dfrac{\\sqrt{2}}{2} \\ .\r\n\\]\r\n\u3067\u3042\u308a, \u3053\u306e\u3068\u304d\r\n\\[\r\nX = 1 \\pm \\sqrt{2} \\ .\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, P \u306e\u8ecc\u8de1\u3068\u76f4\u7dda \\(y=1\\) \u306b\u56f2\u307e\u308c\u308b\u56f3\u5f62\u306f\u4e0b\u56f3\u659c\u7dda\u90e8\u306e\u3088\u3046\u306b\u306a\u308b.<\/p>\r\n\"\"\r\n

    \u3088\u3063\u3066, \u6c42\u3081\u308b\u9762\u7a4d \\(S\\) \u306f\r\n\\[\r\nS = \\displaystyle\\int _ {1-\\sqrt{2}}^{1+\\sqrt{2}} (1-Y) \\, dX \\ .\r\n\\]\r\n\u3053\u3053\u3067, (1)<\/strong> \u306e\u7d50\u679c\u3088\u308a\r\n\\[\r\n\\dfrac{dX}{da} =2(a+1) \\ .\r\n\\]\r\n\u307e\u305f\r\n\\[\r\n\\begin{array}{c|ccc} x & 1-\\sqrt{2} & \\rightarrow & 1+\\sqrt{2} \\\\ \\hline a & -\\dfrac{\\sqrt{2}}{2} & \\rightarrow & \\dfrac{\\sqrt{2}}{2} \\end{array}\r\n\\]\r\n\u306a\u306e\u3067\r\n\\[\\begin{align}\r\nS & = \\displaystyle\\int _ {-\\frac{\\sqrt{2}}{2}}^{\\frac{\\sqrt{2}}{2}} \\left( \\dfrac{1}{2} -a^2 \\right) \\cdot 2(a+1) \\, da \\\\\r\n& = 2 \\displaystyle\\int _ {0}^{\\frac{\\sqrt{2}}{2}} ( 1 -2a^2 ) \\, da \\\\\r\n& = 2 \\left[ a -\\dfrac{2 a^3}{3} \\right] _ {0}^{\\frac{\\sqrt{2}}{2}} \\\\\r\n& = \\sqrt{2} \\left( 1 -\\dfrac{1}{3} \\right) \\\\\r\n& = \\underline{\\dfrac{2 \\sqrt{2}}{3}} \\ .\r\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\\(a\\) \u3092\u5b9f\u6570\u3068\u3059\u308b. \u5186 \\(C\\) \u306f\u70b9 \\((a, -a)\\) \u3067\u76f4\u7dda \\(y = -x\\) \u3092\u63a5\u7dda\u306b\u3082\u3061, \u70b9 \\((0,1)\\) \u3092\u901a\u308b\u3082\u306e\u3068\u3059\u308b. \\(C\\) \u306e\u4e2d\u5fc3\u3092 P \\((X,Y)\\) \u3068\u3057\u3066, … \u7d9a\u304d\u3092\u8aad\u3080 →<\/span><\/a>","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false,"footnotes":""},"categories":[73],"tags":[148,13],"class_list":["post-484","post","type-post","status-publish","format-standard","hentry","category-tohoku_r_2011","tag-tohoku_r","tag-13"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/484","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/comments?post=484"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/484\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=484"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=484"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=484"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}