{"id":136,"date":"2011-11-30T01:17:26","date_gmt":"2011-11-29T16:17:26","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=136"},"modified":"2021-09-14T12:46:03","modified_gmt":"2021-09-14T03:46:03","slug":"iks201003","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/iks201003\/","title":{"rendered":"\u533b\u79d1\u6b6f\u79d1\u59272010\uff1a\u7b2c3\u554f"},"content":{"rendered":"
\n

\\(xy\\) \u5e73\u9762\u306b\u304a\u3044\u3066, \u6b21\u306e\u5186 \\(C\\) \u3068\u6955\u5186 \\(E\\) \u3092\u8003\u3048\u308b.\r\n\\[\\begin{align}\r\nC & : \\ x^2+y^2 = 1 \\\\\r\nE & : \\ x^2+\\dfrac{y^2}{2} = 1\r\n\\end{align}\\]\r\n\u307e\u305f, \\(C\\) \u4e0a\u306e\u70b9 P \\(( s , t )\\) \u306b\u304a\u3051\u308b \\(C\\) \u306e\u63a5\u7dda\u3092 \\(l\\) \u3068\u3059\u308b.\r\n\u3053\u306e\u3068\u304d\u4ee5\u4e0b\u306e\u5404\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n

    \r\n
  1. (1)<\/strong>\u3000\\(l\\) \u306e\u65b9\u7a0b\u5f0f\u3092 \\(s , t\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/li>\r\n<\/ol>\r\n

    \u4ee5\u4e0b, \\(t \\gt 0\\) \u3068\u3057, \\(E\\) \u304c \\(l\\) \u304b\u3089\u5207\u308a\u53d6\u308b\u7dda\u5206\u306e\u9577\u3055\u3092 \\(L\\) \u3068\u3059\u308b.<\/p>\r\n

      \r\n

    1. (2)<\/strong>\u3000\\(L\\) \u3092 \\(t\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n

    2. (3)<\/strong>\u3000P \u304c\u52d5\u304f\u3068\u304d, \\(L\\) \u306e\u6700\u5927\u5024\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

    3. (4)<\/strong>\u3000\\(L\\) \u304c (3)<\/strong> \u3067\u6c42\u3081\u305f\u6700\u5927\u5024\u3092\u3068\u308b\u3068\u304d, \\(l\\) \u3068 \\(E\\) \u304c\u56f2\u3080\u9818\u57df\u306e\u3046\u3061, \u539f\u70b9\u3092\u542b\u307e\u306a\u3044\u9818\u57df\u306e\u9762\u7a4d\u3092 \\(A\\) \u3068\u3059\u308b. \\(A\\) \u306e\u5024\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

      \\[\r\nl : \\ \\underline{sx + ty = 1}\n\\]\r\n

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

      \\(t \\gt 0\\) \u306a\u306e\u3067\r\n\\[\r\nl : \\ y = -\\dfrac{s}{t} x +\\dfrac{1}{t}\n\\]\r\n\\(l\\) \u3068 \\(E\\) \u306e \\(2\\) \u3064\u306e\u4ea4\u70b9\u306e \\(x\\) \u5ea7\u6a19\u3092 \\(\\alpha , \\beta \\ ( \\alpha \\lt \\beta )\\) \u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\nL & = ( \\beta -\\alpha ) \\sqrt{1 + \\left( -\\dfrac{s}{t} \\right)^2} \\\\\r\n& = ( \\beta -\\alpha ) \\sqrt{\\dfrac{t^2 +s^2}{t^2}} \\\\\r\n& = \\dfrac{\\beta -\\alpha}{t} \\quad ... [1] \\quad ( \\ \\text{\u2235} \\ s^2+t^2=1 \\ )\n\\end{align}\\]\r\n\\(l\\) \u3068 \\(E\\) \u306e\u5f0f\u304b\u3089 \\(y\\) \u3092\u6d88\u53bb\u3059\u308b\u3068\r\n\\[\\begin{align}\r\n2t^2x^2 +( 1 -sx )^2 & = 2t^2 \\\\\r\n( 2t^2+s^2 )x^2 -2sx -2t^2+1 & = 0 \\\\\r\n\\text{\u2234} \\quad ( t^2+1 )x^2 -2sx -2t^2+1 & = 0\n\\end{align}\\]\r\n\\(\\alpha , \\beta\\) \u306f\u3053\u306e\u65b9\u7a0b\u5f0f\u306e \\(2\\) \u89e3\u306a\u306e\u3067, \u89e3\u3068\u4fc2\u6570\u306e\u95a2\u4fc2\u3088\u308a\r\n\\[\r\n\\alpha +\\beta = \\dfrac{2s}{t^2+1} , \\ \\alpha \\beta = \\dfrac{-2t^2 +1}{t^2+1}\n\\]\r\n\u3053\u308c\u3092\u7528\u3044\u308b\u3068\r\n\\[\\begin{align}\r\n\\beta -\\alpha & = \\sqrt{( \\beta -\\alpha )^2 -4 \\alpha \\beta} \\\\\r\n& = \\sqrt{ \\left( \\dfrac{2s}{t^2+1} \\right)^2 -4 \\cdot \\dfrac{-2t^2 +1}{t^2+1}} \\\\\r\n& = \\dfrac{2 \\sqrt{(1-t^2) +(2t^2 -1)(t^2+1)}}{t^2+1} \\\\\r\n& = \\dfrac{2 \\sqrt{2} t^2}{t^2+1} \\quad ... [2]\n\\end{align}\\]\r\n[1] [2] \u3088\u308a\r\n\\[\r\nL = \\underline{\\dfrac{2 \\sqrt{2} t}{t^2+1}}\n\\]\r\n

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

      \u76f8\u52a0\u76f8\u4e57\u5e73\u5747\u306e\u95a2\u4fc2\u3092\u7528\u3044\u3066\r\n\\[\\begin{align}\r\n\\dfrac{1}{L} & = \\dfrac{\\sqrt{2}}{4} \\cdot \\left( t +\\dfrac{1}{t} \\right) \\\\\r\n& \\geqq \\dfrac{\\sqrt{2}}{4} \\cdot 2 \\sqrt{t \\cdot \\dfrac{1}{t}} = \\dfrac{\\sqrt{2}}{2}\n\\end{align}\\]\r\n\u7b49\u53f7\u6210\u7acb\u306f, \\(t = \\dfrac{1}{t}\\) \u3059\u306a\u308f\u3061 \\(t=1\\) \u306e\u3068\u304d.
      \r\n\u3086\u3048\u306b \\(L\\) \u306e\u6700\u5927\u5024\u306f\r\n\\[\r\n\\underline{\\sqrt{2}}\n\\]\r\n

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

      \u4e0e\u3048\u3089\u308c\u305f\u9818\u57df\u306f \\(y\\) \u8ef8\u306b\u3064\u3044\u3066\u5bfe\u79f0\u306a\u306e\u3067\r\n\\[\r\nA = 2 \\displaystyle\\int _ 1^{\\sqrt{2}} \\sqrt{1 -\\dfrac{y^2}{2}} \\, dy\n\\]\r\n\\(y = \\sqrt{2} \\sin \\theta\\) \u3068\u304a\u304f\u3068\r\n\\[\\begin{gather}\r\ndy =\\sqrt{2} \\cos \\theta \\, d \\theta , \\\\\r\n\\begin{array}{c|ccc} y & 1 & \\rightarrow & \\sqrt{2} \\\\ \\hline \\theta & \\dfrac{\\pi}{4} & \\rightarrow & \\dfrac{\\pi}{2} \\end{array}\n\\end{gather}\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\\begin{align}\r\nA & = 2 \\displaystyle\\int _ {\\frac{\\pi}{4}}^{\\frac{\\pi}{2}} \\sqrt{1 -\\sin^2 \\theta} \\cdot \\sqrt{2} \\cos \\theta \\, d \\theta \\\\\r\n& = 2 \\sqrt{2} \\displaystyle\\int _ {\\frac{\\pi}{4}}^{\\frac{\\pi}{2}} \\cos^2 \\theta \\, d \\theta \\\\\r\n& = 2 \\sqrt{2} \\displaystyle\\int _ {\\frac{\\pi}{4}}^{\\frac{\\pi}{2}} \\dfrac{1 +\\cos 2\\theta}{2} \\, d \\theta \\\\\r\n& = \\sqrt{2} \\left[ \\theta +\\dfrac{\\sin 2\\theta}{2} \\right] _ {\\frac{\\pi}{4}}^{\\frac{\\pi}{2}} \\\\\r\n& = \\sqrt{2} \\left( \\dfrac{\\pi}{4} -\\dfrac{1}{2} \\right) \\\\\r\n& = \\underline{\\dfrac{\\sqrt{2}}{4} \\left( \\pi -2 \\right)}\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\\(xy\\) \u5e73\u9762\u306b\u304a\u3044\u3066, \u6b21\u306e\u5186 \\(C\\) \u3068\u6955\u5186 \\(E\\) \u3092\u8003\u3048\u308b. \\[\\begin{align} C & : \\ x^2+y^2 = 1 \\\\ E & : \\ x^2+\\dfrac{y^2}{2} = 1 … \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":[28],"tags":[145,14],"class_list":["post-136","post","type-post","status-publish","format-standard","hentry","category-ikashika_2010","tag-ikashika","tag-14"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/136","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=136"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/136\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=136"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=136"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=136"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}