{"id":236,"date":"2011-12-03T00:34:14","date_gmt":"2011-12-02T15:34:14","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=236"},"modified":"2021-10-03T20:33:46","modified_gmt":"2021-10-03T11:33:46","slug":"kbr200906","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/kbr200906\/","title":{"rendered":"\u7b51\u6ce2\u5927\u7406\u7cfb2009\uff1a\u7b2c6\u554f"},"content":{"rendered":"
\n

\u70b9 \\(P \\, ( x, y )\\) \u304c\u53cc\u66f2\u7dda \\(\\dfrac{x^2}{2} -y^2 = 1\\) \u4e0a\u3092\u52d5\u304f\u3068\u304d, \u70b9 \\(P \\, ( x, y )\\) \u3068\u70b9 \\(A \\, ( a, 0 )\\) \u3068\u306e\u8ddd\u96e2\u306e\u6700\u5c0f\u5024\u3092 \\(f(a)\\) \u3068\u3059\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(f(a)\\) \u3092 \\(a\\) \u3067\u8868\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(f(a)\\) \u3092 \\(a\\) \u306e\u95a2\u6570\u3068\u307f\u306a\u3059\u3068\u304d, \\(ab\\) \u5e73\u9762\u4e0a\u306b\u66f2\u7dda \\(b = f(a)\\) \u306e\u6982\u5f62\u3092\u304b\u3051.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

    \u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n

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

    \\(\\dfrac{x^2}{2} -y^2 = 1\\) \u306f, \\(x\\) \u8ef8, \\(y\\) \u8ef8\u305d\u308c\u305e\u308c\u306b\u3064\u3044\u3066\u5bfe\u79f0\u306a\u306e\u3067, \\(x \\geqq \\sqrt{2}\\) , \\(y \\geqq 0\\) , \\(a \\geqq 0\\) \u306e\u5834\u5408\u306b\u3064\u3044\u3066\u8003\u3048\u308b.
    \r\n\u6761\u4ef6\u3088\u308a\r\n\\[\\begin{align}\r\nPA^2 & = (x-a)^2 +y^2 =(x-a)^2 +\\dfrac{x^2}{2} -1 \\\\\r\n& = \\dfrac{3a^2}{2} -2ax +a^2-1 \\\\\r\n& = \\dfrac{3}{2} \\left( x -\\dfrac{2a}{3} \\right)^2 +\\dfrac{a^2}{3}-1\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066<\/p>\r\n

      \r\n

    1. 1*<\/strong>\u3000\\(\\dfrac{2a}{3} \\geqq \\sqrt{2}\\) \u3059\u306a\u308f\u3061 \\(a \\geqq \\dfrac{3 \\sqrt{2}}{2}\\) \u306e\u3068\u304d\r\n\\[\r\nf(a) = \\sqrt{\\dfrac{a^2}{3} -1}\r\n\\]<\/li>\r\n

    2. 2*<\/strong>\u3000\\(\\dfrac{2a}{3} \\lt \\sqrt{2}\\) \u3059\u306a\u308f\u3061 \\(0 \\leqq a \\lt \\dfrac{3 \\sqrt{2}}{2}\\) \u306e\u3068\u304d\r\n\\[\\begin{align}\r\nf(a) & = \\sqrt{\\dfrac{3}{2} \\cdot 2 -2 \\sqrt{2} a +a^2 -1} \\\\\r\n& = \\sqrt{a^2 -2 \\sqrt{2} a +2} =\\sqrt{\\left( a -\\sqrt{2} \\right)^2} \\\\\r\n& = \\left| a -\\sqrt{2} \\right|\r\n\\end{align}\\]<\/li>\r\n<\/ol>\r\n

      1*<\/strong> 2*<\/strong>\u3088\u308a, \\(a \\geqq 0\\) \u306b\u3064\u3044\u3066\r\n\\[\r\nf(a) = \\left\\{ \\begin{array}{ll} \\left| a -\\sqrt{2} \\right| & \\left( \\ 0 \\leqq a \\lt \\dfrac{3 \\sqrt{2}}{2} \\text{\u306e\u3068\u304d} \\right) \\\\ \\sqrt{\\dfrac{a^2}{3} -1} & \\left( \\ \\dfrac{3 \\sqrt{2}}{2} \\leqq a \\text{\u306e\u3068\u304d} \\right) \\end{array} \\right.\r\n\\]\r\n\u3055\u3089\u306b \\(a \\lt 0\\) \u306e\u5834\u5408\u306b\u3064\u3044\u3066\u306f, \\(a\\) \u3092 \\(-a\\) \u306b\u7f6e\u304d\u63db\u3048\u305f\u3082\u306e\u3068\u306a\u308b\u306e\u3067, \u6c42\u3081\u308b\u95a2\u6570 \\(f(a)\\) \u306f\r\n\\[\r\nf(a) = \\underline{\\left\\{ \\begin{array}{ll} \\sqrt{\\dfrac{a^2}{3} -1} & \\left( \\ a \\leqq -\\dfrac{3 \\sqrt{2}}{2} \\text{\u306e\u3068\u304d} \\right) \\\\ \\left| a +\\sqrt{2} \\right| & \\left( \\ -\\dfrac{3 \\sqrt{2}}{2} \\lt a \\lt 0 \\text{\u306e\u3068\u304d} \\right) \\\\ \\left| a -\\sqrt{2} \\right| & \\left( \\ 0 \\leqq a \\lt \\dfrac{3 \\sqrt{2}}{2} \\text{\u306e\u3068\u304d} \\right) \\\\ \\sqrt{\\dfrac{a^2}{3} -1} & \\left( \\ \\dfrac{3 \\sqrt{2}}{2} \\leqq a \\text{\u306e\u3068\u304d} \\right) \\end{array} \\right.}\r\n\\]\r\n

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

      \\(a \\geqq \\dfrac{3 \\sqrt{2}}{2}\\) \u306e\u3068\u304d\r\n\\[\\begin{align}\r\nf(a) -\\dfrac{a}{\\sqrt{3}} & = -\\dfrac{1}{\\sqrt{\\frac{a^2}{3} -1} +\\frac{a}{\\sqrt{3}}} \\\\\r\n& \\rightarrow 0 \\quad ( \\ a \\rightarrow \\infty \\text{\u306e\u3068\u304d} )\r\n\\end{align}\\]\r\n\u5bfe\u79f0\u6027\u3082\u8003\u616e\u3059\u308c\u3070, \\(b = \\pm \\dfrac{a}{\\sqrt{3}}\\) \u306f \\(b = f(a)\\) \u306e\u6f38\u8fd1\u7dda\u3068\u306a\u308b.
      \r\n\u3053\u308c\u3068 (1)<\/strong> \u306e\u7d50\u679c\u3088\u308a, \\(b = f(a)\\) \u306e\u30b0\u30e9\u30d5\u306f\u4e0b\u56f3\u306e\u3088\u3046\u306b\u306a\u308b.<\/p>\r\n\"\"\r\n","protected":false},"excerpt":{"rendered":"\u70b9 \\(P \\, ( x, y )\\) \u304c\u53cc\u66f2\u7dda \\(\\dfrac{x^2}{2} -y^2 = 1\\) \u4e0a\u3092\u52d5\u304f\u3068\u304d, \u70b9 \\(P \\, ( x, y )\\) \u3068\u70b9 \\(A \\, ( a, 0 )\\) \u3068\u306e\u8ddd\u96e2\u306e\u6700\u5c0f\u5024 … \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":[39],"tags":[144,15],"class_list":["post-236","post","type-post","status-publish","format-standard","hentry","category-tsukuba_r_2009","tag-tsukuba_r","tag-15"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/236","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=236"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/236\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=236"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=236"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=236"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}