{"id":104,"date":"2011-11-27T19:00:02","date_gmt":"2011-11-27T10:00:02","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=104"},"modified":"2021-03-18T10:05:17","modified_gmt":"2021-03-18T01:05:17","slug":"tkr201006","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tkr201006\/","title":{"rendered":"\u6771\u5927\u7406\u7cfb2010\uff1a\u7b2c6\u554f"},"content":{"rendered":"
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\u56db\u9762\u4f53 OABC \u306b\u304a\u3044\u3066, \\(4\\) \u3064\u306e\u9762\u306f\u3059\u3079\u3066\u5408\u540c\u3067\u3042\u308a, \\(\\text{OA} = 3\\) , \\(\\text{OB} = \\sqrt{7}\\) , \\(\\text{OC} = 2\\) \u3067\u3042\u308b\u3068\u3059\u308b. \u307e\u305f \\(3\\) \u70b9 O , A , B \u3092\u542b\u3080\u5e73\u9762\u3092 \\(L\\) \u3068\u3059\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\u70b9 C \u304b\u3089\u5e73\u9762 \\(L\\) \u306b\u304a\u308d\u3057\u305f\u5782\u7dda\u306e\u8db3\u3092 H \u3068\u304a\u304f.\r\n\\(\\overrightarrow{\\text{OH}}\\) \u3092 \\(\\overrightarrow{\\text{OA}}\\) \u3068 \\(\\overrightarrow{\\text{OB}}\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(0 \\lt t \\lt 1\\) \u3092\u307f\u305f\u3059\u5b9f\u6570 \\(t\\) \u306b\u5bfe\u3057\u3066,\r\n\u7dda\u5206 OA , OB \u306e\u5404\u3005\u3092 \\(t : (1-t)\\) \u306b\u5185\u5206\u3059\u308b\u70b9\u3092\u305d\u308c\u305e\u308c \\(\\text{P} {} _ t , \\text{Q} {} _ t\\) \u3092\u901a\u308a,\r\n\u5e73\u9762 \\(L\\) \u306b\u5782\u76f4\u306a\u5e73\u9762\u3092 \\(M\\) \u3068\u3059\u308b\u3068\u304d, \u5e73\u9762 \\(M\\) \u306b\u3088\u308b\u56db\u9762\u4f53 OABC \u306e\u5207\u308a\u53e3\u306e\u9762\u7a4d \\(S(t)\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\\(t\\) \u304c \\(0 \\lt t \\lt 1\\) \u306e\u7bc4\u56f2\u3092\u52d5\u304f\u3068\u304d, \\(S(t)\\) \u306e\u6700\u5927\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

    \\(\\overrightarrow{\\text{OA}} = \\overrightarrow{a}\\) , \\(\\overrightarrow{\\text{OB}} = \\overrightarrow{b}\\) , \\(\\overrightarrow{\\text{OC}} = \\overrightarrow{c}\\) \u3068\u304a\u304f.
    \r\n\u6761\u4ef6\u3088\u308a\r\n\\[\r\n\\text{OA} = \\text{BC} = 3 , \\ \\text{OB} = \\text{CA} = \\sqrt{7} , \\ \\text{OC} = \\text{AB} = 2\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066,\u3000\\(\\left| \\overrightarrow{a} \\right| = 3\\) , \\(\\left| \\overrightarrow{b} \\right| = \\sqrt{7}\\) , \\(\\left| \\overrightarrow{c} \\right| = 2\\) .
    \r\n\u3055\u3089\u306b\r\n\\[\\begin{align}\r\n\\left| \\overrightarrow{\\text{BC}} \\right|^2 & = \\left| \\overrightarrow{c} \\right|^2 -2 \\overrightarrow{b} \\cdot \\overrightarrow{c} +\\left| \\overrightarrow{b} \\right|^2 \\\\\r\n& = 4 -2\\overrightarrow{b} \\cdot \\overrightarrow{c} +7 = 9 \\\\\r\n\\text{\u2234} & \\quad \\overrightarrow{b} \\cdot \\overrightarrow{c} = 1\r\n\\end{align}\\]\r\n\\[\\begin{align}\r\n\\left| \\overrightarrow{\\text{CA}} \\right|^2 & = \\left| \\overrightarrow{a} \\right|^2 -2 \\overrightarrow{c} \\cdot \\overrightarrow{a} +\\left| \\overrightarrow{c} \\right|^2 \\\\\r\n& = 9 -2\\overrightarrow{c} \\cdot \\overrightarrow{a} +4 = 7 \\\\\r\n\\text{\u2234} & \\quad \\overrightarrow{c} \\cdot \\overrightarrow{a} = 3\r\n\\end{align}\\]\r\n\\[\\begin{align}\r\n\\left| \\overrightarrow{\\text{AB}} \\right|^2 & = \\left| \\overrightarrow{b} \\right|^2 -2 \\overrightarrow{a} \\cdot \\overrightarrow{b} +\\left| \\overrightarrow{a} \\right|^2 \\\\\r\n& = 7 -2\\overrightarrow{a} \\cdot \\overrightarrow{b} +9 = 4 \\\\\r\n\\text{\u2234} & \\quad \\overrightarrow{b} \\cdot \\overrightarrow{c} = 6\r\n\\end{align}\\]\r\n\\(\\overrightarrow{\\text{OH}} = x \\overrightarrow{a} +y \\overrightarrow{b}\\) \u3068\u304a\u304f\u3068, \\(\\text{CA} \\perp \\text{OA}\\) , \\(\\text{CA} \\perp \\text{OA}\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\n\\overrightarrow{\\text{CH}} \\cdot \\overrightarrow{\\text{OA}} & = \\left( x \\overrightarrow{a} +y \\overrightarrow{b} -\\overrightarrow{c} \\right) \\cdot \\overrightarrow{a} \\\\\r\n& = 9x +6y -3 = 0 \\\\\r\n\\text{\u2234} & \\quad 3x +2y = 1 \\ ... [1]\r\n\\end{align}\\]\r\n\\[\\begin{align}\r\n\\overrightarrow{\\text{CH}} \\cdot \\overrightarrow{\\text{OB}} & = \\left( x \\overrightarrow{a} +y \\overrightarrow{b} -\\overrightarrow{c} \\right) \\cdot \\overrightarrow{b} \\\\\r\n& = 6x +7y -1 = 0 \\\\\r\n\\text{\u2234} & \\quad 6x +7y = 1 \\ ... [2]\r\n\\end{align}\\]\r\n[1] [2] \u3088\u308a\r\n\\[\r\nx = \\dfrac{5}{9}, \\quad y = -\\dfrac{1}{3}\r\n\\]\r\n\u3088\u3063\u3066\r\n\\[\r\n\\overrightarrow{\\text{OH}} = \\underline{\\dfrac{5}{9} \\overrightarrow{\\text{OA}} -\\dfrac{1}{3} \\overrightarrow{\\text{OB}}}\r\n\\]\r\n

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

    (1)<\/strong> \u306e\u7d50\u679c\u3088\u308a\r\n\\[\r\n\\overrightarrow{\\text{OH}} = \\dfrac{5\\overrightarrow{a} -3\\overrightarrow{b}}{9} = \\dfrac{2}{9} \\cdot \\dfrac{5\\overrightarrow{a} -3\\overrightarrow{b}}{2}\r\n\\]\r\nBA \u3092 \\(5 : 3\\) \u306b\u5916\u5206\u3059\u308b\u70b9\u3092 D \u3068\u3059\u308c\u3070, H \u306f OD \u3092 \\(2 : 7\\) \u306b\u5185\u5206\u3059\u308b\u70b9\u3068\u306a\u308b.<\/p>\r\n\"\"\r\n

      \r\n

    1. 1*<\/strong>\u3000\\(t = \\dfrac{2}{9}\\) \u306e\u3068\u304d\r\n\\[\\begin{align}\r\n\\left| \\overrightarrow{\\text{CH}} \\right|^2 & = \\left| \\dfrac{5}{9} \\overrightarrow{a} -\\dfrac{1}{3} \\overrightarrow{b} -\\overrightarrow{c} \\right|^2 \\\\\r\n& = \\dfrac{25}{81} \\cdot 9 +\\dfrac{1}{9} \\cdot 7 +4 -2 \\cdot \\dfrac{5}{27} \\cdot 6 -2 \\cdot \\dfrac{5}{9} \\cdot 3 +2 \\cdot \\dfrac{1}{3} \\cdot 1 \\\\\r\n& = \\dfrac{25}{9} +\\dfrac{7}{9} +4 -\\dfrac{20}{9} -\\dfrac{30}{9} +\\dfrac{2}{3} \\\\\r\n& = \\dfrac{8}{3} \\\\\r\n\\text{\u2234} & \\quad \\text{CH} = \\dfrac{2 \\sqrt{6}}{3}\r\n\\end{align}\\]\r\n\u307e\u305f\r\n\\[\r\n\\text{P} _ {\\frac{2}{9}}\\text{Q} _ {\\frac{2}{9}} = 2 \\cdot \\dfrac{2}{9} = \\dfrac{4}{9}\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\r\nS\\left( \\dfrac{2}{9} \\right) = \\dfrac{1}{2} \\cdot \\dfrac{4}{9} \\cdot \\dfrac{2 \\sqrt{6}}{3} = \\dfrac{4 \\sqrt{6}}{27}\r\n\\]<\/li>\r\n

    2. 2*<\/strong>\u3000\\(0 \\lt t \\lt \\dfrac{2}{9}\\) \u306e\u3068\u304d
      \r\n\u5207\u308a\u53e3\u3068 OC \u306e\u4ea4\u70b9 \\(\\text{R} {} _ t\\) \u306f,\r\nOC \u3092 \\(t : \\left( \\dfrac{2}{9} -t \\right) = 9t : (2-9t)\\) \u306b\u5185\u5206\u3059\u308b.
      \r\n\u5207\u308a\u53e3 \\(\\text{P} {} _ t\\text{Q} {} _ t\\text{R} {} _ t\\) \u306f\u4e09\u89d2\u5f62\u3067\r\n\\[\\begin{align}\r\n\\text{P} _ t\\text{Q} _ t & = t \\text{AB} = 2t \\\\\r\n( \\text{\u9ad8\u3055} ) & = \\dfrac{9t}{2} \\text{CH} = 3\\sqrt{6} t\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\r\nS(t) = \\dfrac{1}{2} \\cdot 2t \\cdot 3\\sqrt{6} t = 3\\sqrt{6} t^2\r\n\\]<\/li>\r\n

    3. 3*<\/strong>\u3000\\(\\dfrac{2}{9} \\lt t \\lt 1\\) \u306e\u3068\u304d
      \r\n\u5207\u308a\u53e3\u3068 BC , AC \u306e\u4ea4\u70b9 \\(\\text{S} {} _ t\\) , \\(\\text{T} {} _ t\\) \u306f,\r\n\u305d\u308c\u305e\u308c\u3092 \\(\\left( t -\\dfrac{2}{9} \\right) : (1-t) = (9t-2) : 9(1-t)\\) \u306b\u5185\u5206\u3059\u308b.
      \r\n\u5207\u308a\u53e3 \\(\\text{P} {} _ t \\text{Q} {} _ t \\text{S} {} _ t \\text{T} {} _ t\\) \u306f\u53f0\u5f62\u3067,\r\n\\[\\begin{align}\r\n\\text{P} {} _ t \\text{Q} {} _ t & = 2t \\\\\r\n\\text{S} {} _ t \\text{T} {} _ t & = \\dfrac{9t-2}{7} \\text{AB} = \\dfrac{2}{7} ( 9t-2 ) \\\\\r\n( \\text{\u9ad8\u3055} ) & = \\left( 1 -\\dfrac{9t-2}{7} \\right) \\text{CH} = \\dfrac{6 \\sqrt{6}}{7} ( 1-t )\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\\begin{align}\r\nS(t) & = \\dfrac{1}{2} \\cdot \\left\\{ 2t +\\dfrac{2}{7} ( 9t-2 ) \\right\\} \\cdot \\dfrac{6 \\sqrt{6}}{7} ( 1-t ) \\\\\r\n& = \\dfrac{4}{7} ( 8t-1 ) \\cdot \\dfrac{3 \\sqrt{6}}{7} ( 1-t ) \\\\\r\n& = \\dfrac{12 \\sqrt{6}}{49} ( 8t-1 ) ( 1-t )\r\n\\end{align}\\]<\/li>\r\n<\/ol>\r\n

      \u4ee5\u4e0a\u3088\u308a\r\n\\[\r\nS(t) = \\underline{\\left\\{ \\begin{array}{ll} 3\\sqrt{6} t^2 & \\ \\left( 0 \\lt t \\leqq \\dfrac{2}{9} \\text{\u306e\u3068\u304d} \\right) \\\\ \\dfrac{12 \\sqrt{6}}{49} ( 8t-1 ) ( 1-t ) & \\ \\left( \\dfrac{2}{9} \\lt t \\lt 1 \\text{\u306e\u3068\u304d} \\right) \\end{array} \\right. }\r\n\\]\r\n

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

      \\(S(t) = \\dfrac{12 \\sqrt{6}}{49} f(x)\\) \u3068\u306a\u308b \\(f(x)\\) \u306b\u3064\u3044\u3066\u5897\u6e1b\u3092\u8003\u3048\u308c\u3070\u3088\u3044.<\/p>\r\n

        \r\n

      1. 1*<\/strong>\u3000\\(0 \\lt t \\leqq \\dfrac{2}{9}\\) \u306e\u3068\u304d\r\n\\[\r\nf(t) = \\dfrac{49}{4} t^2\r\n\\]<\/li>\r\n

      2. 2*<\/strong>\u3000\\(\\dfrac{2}{9} \\lt t \\lt 1\\) \u306e\u3068\u304d\r\n\\[\\begin{align}\r\nf(t) & = ( 8t-1 ) ( 1-t ) \\\\\r\n& = -8t^2 +9t -1 \\\\\r\n& = -8 \\left( t -\\dfrac{9}{16} \\right)^2 +\\dfrac{49}{32}\r\n\\end{align}\\]<\/li>\r\n<\/ol>\r\n

        \u4ee5\u4e0a\u3088\u308a \\(0 \\lt t \\lt 1\\) \u306b\u304a\u3051\u308b \\(f(x)\\) \u306e\u30b0\u30e9\u30d5\u306f\u4e0b\u56f3.<\/p>\r\n\"\"\r\n

        \u3057\u305f\u304c\u3063\u3066, \\(f(x)\\) \u306e\u6700\u5927\u5024\u306f\r\n\\[\r\nf \\left( \\dfrac{9}{16} \\right) = \\dfrac{49}{32}\r\n\\]\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u6700\u5927\u5024\u306f\r\n\\[\r\nS \\left( \\dfrac{9}{16} \\right) = \\dfrac{12 \\sqrt{6}}{49} \\cdot \\dfrac{49}{32} = \\underline{\\dfrac{3 \\sqrt{6}}{8}}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u56db\u9762\u4f53 OABC \u306b\u304a\u3044\u3066, \\(4\\) \u3064\u306e\u9762\u306f\u3059\u3079\u3066\u5408\u540c\u3067\u3042\u308a, \\(\\text{OA} = 3\\) , \\(\\text{OB} = \\sqrt{7}\\) , \\(\\text{OC} = 2\\) \u3067\u3042\u308b\u3068\u3059\u308b. \u307e\u305f … \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":[19],"tags":[139,14],"class_list":["post-104","post","type-post","status-publish","format-standard","hentry","category-tokyo_r_2010","tag-tokyo_r","tag-14"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/104","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=104"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/104\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=104"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=104"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=104"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}