{"id":1020,"date":"2015-04-07T20:00:07","date_gmt":"2015-04-07T11:00:07","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=1020"},"modified":"2021-11-04T08:55:59","modified_gmt":"2021-11-03T23:55:59","slug":"htb201303","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/htb201303\/","title":{"rendered":"\u4e00\u6a4b\u59272013\uff1a\u7b2c3\u554f"},"content":{"rendered":"
\n

\u539f\u70b9\u3092 O \u3068\u3059\u308b \\(xy\\) \u5e73\u9762\u4e0a\u306b, \u653e\u7269\u7dda \\(C\\) \uff1a \\(y = 1-x^2\\) \u304c\u3042\u308b.\r\n\\(C\\) \u4e0a\u306b \\(2\\) \u70b9 P \\(( p , 1-p^2 )\\) , Q \\(( q , 1-q^2 )\\) \u3092 \\(p \\lt q\\) \u3068\u306a\u308b\u3088\u3046\u306b\u3068\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(2\\) \u3064\u306e\u7dda\u5206 OP , OQ \u3068\u653e\u7269\u7dda \\(C\\) \u3067\u56f2\u307e\u308c\u305f\u90e8\u5206\u306e\u9762\u7a4d \\(S\\) \u3092, \\(p\\) \u3068 \\(q\\) \u306e\u5f0f\u3067\u8868\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(q = p+1\\) \u3067\u3042\u308b\u3068\u304d \\(S\\) \u306e\u6700\u5c0f\u5024\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\\(pq = -1\\) \u3067\u3042\u308b\u3068\u304d \\(S\\) \u306e\u6700\u5c0f\u5024\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

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

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

    \u76f4\u7dda PQ \u306e\u5f0f\u306f\r\n\\[\\begin{align}\r\ny & = \\dfrac{(1-q^2) -(1-p^2)}{q-p} (x-p) +1-p^2 \\\\\r\n& = -(p+q) (x-p) +1-p^2 \\\\\r\n& = -(p+q) x +1+pq\r\n\\end{align}\\]\r\n\u7dda\u5206 PQ \u3068\u653e\u7269\u7dda \\(C\\) \u306b\u56f2\u307e\u308c\u308b\u90e8\u5206 \\(D\\) \u306e\u9762\u7a4d\u3092 \\(S _ 1\\) , \u25b3OPQ \u306e\u9762\u7a4d\u3092 \\(S _ 2\\) \u3068\u304a\u304f.
    \r\n\\(S _ 1\\) \u306b\u3064\u3044\u3066\r\n\\[\\begin{align}\r\nS _ 1 & = \\displaystyle\\int _ {p}^{q} \\left[ (1-x^2) -\\left\\{ -(p+q) x +1-pq \\right\\} \\right] \\, dx \\\\\r\n& = -\\displaystyle\\int _ {p}^{q} (x-p)(x-q) \\, dx \\\\\r\n& = \\dfrac{1}{6} (q-p)^3\r\n\\end{align}\\]\r\n\\(S _ 2\\) \u306b\u3064\u3044\u3066\r\n\\[\\begin{align}\r\nS _ 2 & = \\dfrac{1}{2} \\left| p(1-q^2) -q(1-p^2) \\right| \\\\\r\n& = \\dfrac{1}{2} \\left| p-q +pq(p-q) \\right| \\\\\r\n& = \\dfrac{1}{2} (q-p) \\left| 1+pq \\right|\r\n\\end{align}\\]\r\n\u3053\u3053\u3067, \u70b9 O \u3068\u7dda\u5206 PQ \u306e\u4f4d\u7f6e\u95a2\u4fc2\u3067\u5834\u5408\u5206\u3051\u3057\u3066\u8003\u3048\u308b.<\/p>\r\n

      \r\n

    1. 1*<\/strong>\u3000O \u304c PQ \u3088\u308a\u4e0a\u5074\u306b\u3042\u308b, \u3059\u306a\u308f\u3061 \\(1+pq \\geqq 0\\) \u306e\u3068\u304d\r\n\\(S _ 2 = \\dfrac{1}{2} (q-p)(1+pq)\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\nS & = S _ 1 +S _ 2 \\\\\r\n& = \\dfrac{1}{6} (q-p)^3 +\\dfrac{1}{2} (q-p)(1+pq)\r\n\\end{align}\\]<\/li>\r\n

    2. 2*<\/strong>\u3000O \u304c PQ \u3088\u308a\u4e0b\u5074\u306b\u3042\u308b, \u3059\u306a\u308f\u3061 \\(1+pq \\lt 0\\) \u306e\u3068\u304d\r\n\\(S _ 2 = -\\dfrac{1}{2} (q-p)(1+pq)\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\nS & = S _ 1 -S _ 2 \\\\\r\n& = \\dfrac{1}{6} (q-p)^3 +\\dfrac{1}{2} (q-p)(1+pq)\r\n\\end{align}\\]<\/li>\r\n<\/ol>\r\n

      \u4ee5\u4e0a\u3088\u308a, \u3044\u305a\u308c\u306e\u5834\u5408\u3082\u6c42\u3081\u308b\u9762\u7a4d \\(S\\) \u306f\r\n\\[\\begin{align}\r\nS & = \\dfrac{1}{6} (q-p)^3 +\\dfrac{1}{2} (q-p)(1+pq) \\\\\r\n& = \\dfrac{1}{6} (q-p) \\left\\{ (q-p)^2 +3(1+pq) \\right\\} \\\\\r\n& = \\underline{\\dfrac{1}{6} (q-p)( p^2 +pq +q^2 +3 )}\r\n\\end{align}\\]\r\n

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

      \u6761\u4ef6\u3092 (1)<\/strong> \u306e\u7d50\u679c\u306b\u4ee3\u5165\u3059\u308c\u3070\r\n\\[\\begin{align}\r\nS & = \\dfrac{1}{6} +\\dfrac{1}{2} \\left\\{ 1 +p(p+1) \\right\\} \\\\\r\n& = \\dfrac{1}{2} p^2 +\\dfrac{1}{2} p +\\dfrac{2}{3} \\\\\r\n& = \\dfrac{1}{2} \\left( p +\\dfrac{1}{2} \\right)^2 +\\dfrac{13}{24}\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, \\(S\\) \u306f \\(p = -\\dfrac{1}{2}\\) , \\(q = \\dfrac{1}{2}\\) \u306e\u3068\u304d, \u6700\u5c0f\u5024 \\(\\underline{\\dfrac{13}{24}}\\) \u3092\u3068\u308b.<\/p>\r\n

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

      \u6761\u4ef6\u3088\u308a, \\(S _ 2 = 0\\) \u306a\u306e\u3067\r\n\\[\r\nS = \\dfrac{1}{6} \\left( q +\\dfrac{1}{q} \\right)^3\r\n\\]\r\n\\(pq \\lt 0\\) \u3088\u308a, \\(q \\gt 0\\) \u3060\u304b\u3089, \u76f8\u52a0\u76f8\u4e57\u5e73\u5747\u306e\u95a2\u4fc2\u3088\u308a\r\n\\[\r\nq +\\dfrac{1}{q} \\geqq 2 \\sqrt{ q \\cdot \\dfrac{1}{q} } = 2\r\n\\]\r\n\u7b49\u53f7\u6210\u7acb\u306f, \\(q = \\dfrac{1}{q}\\) \u3059\u306a\u308f\u3061 \\(q = 1\\) \u306e\u3068\u304d.
      \r\n\u3088\u3063\u3066, \\(S\\) \u306f, \\(p = -1\\) , \\(q = 1\\) \u306e\u3068\u304d, \u6700\u5c0f\u5024 \\(\\dfrac{1}{6} \\cdot 8 = \\underline{\\dfrac{4}{3}}\\) \u3092\u3068\u308b.<\/p>\r\n","protected":false},"excerpt":{"rendered":"\u539f\u70b9\u3092 O \u3068\u3059\u308b \\(xy\\) \u5e73\u9762\u4e0a\u306b, \u653e\u7269\u7dda \\(C\\) \uff1a \\(y = 1-x^2\\) \u304c\u3042\u308b. \\(C\\) \u4e0a\u306b \\(2\\) \u70b9 P \\(( p , 1-p^2 )\\) , Q \\(( q , 1-q^2 ) … \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":[108],"tags":[146,111],"class_list":["post-1020","post","type-post","status-publish","format-standard","hentry","category-hitotsubashi_2013","tag-hitotsubashi","tag-111"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1020","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=1020"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1020\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1020"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1020"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1020"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}