{"id":239,"date":"2011-12-03T00:38:41","date_gmt":"2011-12-02T15:38:41","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=239"},"modified":"2021-10-20T16:30:36","modified_gmt":"2021-10-20T07:30:36","slug":"ykr200902","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/ykr200902\/","title":{"rendered":"\u6a2a\u56fd\u5927\u7406\u7cfb2009\uff1a\u7b2c2\u554f"},"content":{"rendered":"
\n

\\(xy\\) \u5e73\u9762\u4e0a\u306b\u66f2\u7dda \\(C : \\ y = x^2\\) \u304c\u3042\u308b.\r\n\\(C\\) \u4e0a\u306e\u70b9 P \\(( t , t^2 )\\) \u3092\u6b21\u306e\u6761\u4ef6 (\uff0a) \u3092\u307f\u305f\u3059\u3088\u3046\u306b\u3068\u308b.<\/p>\r\n

    \r\n
  1. (\uff0a)\u3000P \u4ee5\u5916\u306e \\(C\\) \u4e0a\u306e\u7570\u306a\u308b \\(2\\) \u70b9 Q , R \u304c\u3042\u308a, \u305d\u3053\u3067\u306e \\(C\\) \u306e\u6cd5\u7dda\u304c\u3068\u3082\u306b P \u3092\u901a\u308b.<\/li>\r\n<\/ol>\r\n

    \\(\\text{Q} \\ ( \\alpha , \\alpha^2 ) , \\ \\text{R} \\ ( \\beta , \\beta^2 ) \\quad ( \\alpha \\lt \\beta )\\) \u3068\u3059\u308b\u3068\u304d, \u6b21\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n

      \r\n

    1. (1)<\/strong>\u3000\\(t\\) \u306e\u53d6\u308a\u5f97\u308b\u5024\u306e\u7bc4\u56f2\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

    2. (2)<\/strong>\u3000\\(t\\) \u304c (1)<\/strong> \u3067\u6c42\u3081\u305f\u7bc4\u56f2\u3092\u52d5\u304f\u3068\u304d, \u7dda\u5206 QR \u306e\u4e2d\u70b9 M \u304c\u63cf\u304f\u8ecc\u8de1\u306e\u65b9\u7a0b\u5f0f\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

    3. (3)<\/strong>\u3000\\(\\beta\\) \u3092 \\(t\\) \u306e\u5f0f\u3067\u8868\u3057, \u6975\u9650 \\(\\displaystyle\\lim _ {t \\rightarrow \\infty} t \\beta\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n

      \r\n\r\n

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

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

      \u70b9 \\(( p , p^2 ) \\ ( p \\neq t )\\) \u306b\u304a\u3051\u308b \\(C\\) \u306e\u6cd5\u7dda\u306e\u5f0f\u306f\r\n\\[\r\ny = -\\dfrac{1}{2p} (x-p) +p^2\r\n\\]\r\n\u3053\u308c\u304c\u70b9 \\(( t , t^2 )\\) \u3092\u901a\u308b\u306e\u3067\r\n\\[\\begin{align}\r\nt^2 =-\\dfrac{t-p}{2p} & +p^2 \\\\\r\n2p (t-p)(t+p) & = -(t-p) \\\\\r\n\\text{\u2234} \\quad ( t-p )( \\underline{2p^2 +2tp +1} ) & = 0\r\n\\end{align}\\]\r\n\u3053\u308c\u304c, \\(p \\neq t\\) \u3067\u3042\u308b\u7570\u306a\u308b \\(2\\) \u3064\u306e\u5b9f\u6570\u89e3\u3092\u3082\u3066\u3070\u3088\u3044.
      \r\n\u4e0b\u7dda\u90e8\u3092 \\(f(p)\\) \u3068\u304a\u304f\u3068,\r\n\\[\r\nf(t) = 4t^2 +1 \\gt 0\r\n\\]\r\n\u306a\u306e\u3067, \\(f(p) =0\\) \u306f \\(p = t\\) \u3092\u89e3\u306b\u3082\u3064\u3053\u3068\u306f\u306a\u3044\u305f\u3081, \\(f(p) = 0\\) \u304c\u7570\u306a\u308b \\(2\\) \u3064\u306e\u5b9f\u6570\u89e3\u3092\u6301\u3066\u3070\u3088\u304f, \u5224\u5225\u5f0f \\(D\\) \u306b\u3064\u3044\u3066\r\n\\[\\begin{align}\r\n\\dfrac{D}{4} & = t^2 -2 >0 \\\\\r\n\\text{\u2234} \\quad & \\underline{t \\lt -\\sqrt{2} , \\sqrt{2} \\lt t}\r\n\\end{align}\\]\r\n

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

      \\(\\alpha , \\beta\\) \u306f \\(f(p) =0\\) \u306e \\(2\\) \u89e3\u306a\u306e\u3067, \u89e3\u3068\u4fc2\u6570\u306e\u95a2\u4fc2\u3088\u308a\r\n\\[\r\n\\alpha +\\beta = -t , \\ \\alpha \\beta = \\dfrac{1}{2}\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, PQ \u306e\u4e2d\u70b9 M \\(( X , Y )\\) \u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\nX = \\dfrac{\\alpha +\\beta}{2} & = -\\dfrac{t}{2} \\\\\r\n\\text{\u2234} \\quad t & = -2X\r\n\\end{align}\\]\r\n\u3053\u308c\u3092\u7528\u3044\u308c\u3070\r\n\\[\\begin{align}\r\nY & = \\dfrac{\\alpha^2 +\\beta^2}{2} \\\\\r\n& = \\dfrac{(-t)^2 -2 \\cdot \\frac{1}{2}}{2} \\\\\r\n& = 2X^2 -\\dfrac{1}{2}\r\n\\end{align}\\]\r\n\u307e\u305f, (1)<\/strong> \u306e\u7d50\u679c\u3088\u308a\r\n\\[\\begin{align}\r\n-2X \\lt -\\sqrt{2} & , \\sqrt{2} \\lt -2X \\\\\r\n\\text{\u2234} \\quad -\\dfrac{\\sqrt{2}}{2} & \\lt X \\lt \\dfrac{\\sqrt{2}}{2}\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u8ecc\u8de1\u306e\u65b9\u7a0b\u5f0f\u306f\r\n\\[\r\n\\underline{y = 2x^2 -\\dfrac{1}{2} \\quad \\left( -\\dfrac{\\sqrt{2}}{2} \\lt x \\lt \\dfrac{\\sqrt{2}}{2} \\right)}\r\n\\]\r\n

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

      \\[\r\n\\beta = \\underline{\\dfrac{-t +\\sqrt{t^2 -2}}{2}}\r\n\\]\r\n\u306a\u306e\u3067\r\n\\[\\begin{align}\r\nt \\beta & = \\dfrac{t \\left( \\sqrt{t^2 -2} -t \\right)}{2} \\\\\r\n& = \\dfrac{-2t}{2 \\left( t +\\sqrt{t^2 -2} \\right)} \\\\\r\n& = -\\dfrac{1}{1 +\\sqrt{1 +\\frac{2}{t^2}}} \\\\\r\n& \\rightarrow -\\dfrac{1}{2} \\quad ( \\ t \\rightarrow \\infty \\text{\u306e\u3068\u304d} )\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\r\n\\displaystyle\\lim _ {t \\rightarrow \\infty} t \\beta = \\underline{-\\dfrac{1}{2}}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(xy\\) \u5e73\u9762\u4e0a\u306b\u66f2\u7dda \\(C : \\ y = x^2\\) \u304c\u3042\u308b. \\(C\\) \u4e0a\u306e\u70b9 P \\(( t , t^2 )\\) \u3092\u6b21\u306e\u6761\u4ef6 (\uff0a) \u3092\u307f\u305f\u3059\u3088\u3046\u306b\u3068\u308b. (\uff0a)\u3000P \u4ee5\u5916\u306e \\(C\\) \u4e0a\u306e\u7570\u306a\u308b \\( … \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":[42],"tags":[15,9],"class_list":["post-239","post","type-post","status-publish","format-standard","hentry","category-yokokoku_r_2009","tag-15","tag-yokokoku_r"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/239","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=239"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/239\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=239"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=239"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=239"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}