{"id":243,"date":"2011-12-03T00:44:37","date_gmt":"2011-12-02T15:44:37","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=243"},"modified":"2021-10-20T16:34:40","modified_gmt":"2021-10-20T07:34:40","slug":"ykr200905","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/ykr200905\/","title":{"rendered":"\u6a2a\u56fd\u5927\u7406\u7cfb2009\uff1a\u7b2c5\u554f"},"content":{"rendered":"
\n

\u5e73\u9762\u4e0a\u306b \\(3\\) \u70b9 O , A , B \u304c\u3042\u308a, \\(\\text{OA} = a , \\ \\text{OB} = b \\ ( 0 \\lt a \\lt b )\\) \u3067, \\(\\overrightarrow{\\text{OA}}\\) \u3068 \\(\\overrightarrow{\\text{OB}}\\) \u306e\u306a\u3059\u89d2 \\(\\theta\\) \u306f \\(0 \\lt \\theta \\leqq \\dfrac{\\pi}{2}\\) \u3092\u307f\u305f\u3059.\r\n\u70b9 C \u3092 \\(\\overrightarrow{\\text{OC}} = \\overrightarrow{\\text{OA}} +\\overrightarrow{\\text{OB}}\\) \u3067\u5b9a\u3081\u308b.\r\n\u307e\u305f, O \u304b\u3089\u5f15\u3044\u305f\u534a\u76f4\u7dda OA \u4e0a\u306b, \u70b9 P \u3092 \\(\\text{OA} \\lt \\text{OP}\\) \u3068\u306a\u308b\u3088\u3046\u306b\u3068\u308b.\r\n\u76f4\u7dda PC \u3068\u76f4\u7dda OB \u306e\u4ea4\u70b9\u3092 Q \u3068\u3059\u308b.\r\n\\(\\text{AP} = x\\) , \\(\\text{PQ}^2 = f(x)\\) \u3068\u3059\u308b\u3068\u304d, \u6b21\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(f(x)\\) \u3092 \\(x\\) \u3068 \\(a , b , \\theta\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\u7b2c \\(2\\) \u6b21\u5c0e\u95a2\u6570 \\(f''(x)\\) \u306f, \\(x \\gt 0\\) \u306e\u3068\u304d \\(f''(x) \\gt 0\\) \u3092\u307f\u305f\u3059\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\\(a = 1\\) , \\(b = \\sqrt{6}\\) , \\(\\cos \\theta = \\dfrac{1}{\\sqrt{6}}\\) \u306e\u3068\u304d, PQ \u306e\u9577\u3055\u306e\u6700\u5c0f\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}\\) \u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\n\\overrightarrow{\\text{OP}} & = \\dfrac{x+a}{a} \\overrightarrow{a} , \\\\\r\n\\overrightarrow{\\text{OC}} & = \\overrightarrow{a} +\\overrightarrow{b}\r\n\\end{align}\\]\r\n\u70b9 Q \u306f\u76f4\u7dda PC \u4e0a\u306a\u306e\u3067, \u5b9f\u6570 \\(t\\) \u3092\u7528\u3044\u3066\r\n\\[\\begin{align}\r\n\\overrightarrow{\\text{OQ}} & = (1-t) \\overrightarrow{\\text{OP}} +t \\overrightarrow{\\text{OC}} \\\\\r\n& = \\left\\{ \\dfrac{(1-t)(x+a)}{a} +t \\right\\} \\overrightarrow{a} +t \\overrightarrow{b} \\\\\r\n& = \\dfrac{x+a-tx}{a} \\overrightarrow{a} +t \\overrightarrow{b}\r\n\\end{align}\\]\r\n\u3055\u3089\u306b\u70b9 Q \u306f\u76f4\u7dda OB \u4e0a\u306e\u70b9\u306a\u306e\u3067\r\n\\[\\begin{gather}\r\n\\dfrac{x+a-tx}{a} & = 0 \\\\\r\n\\text{\u2234} \\quad t = \\dfrac{x+a}{x}\r\n\\end{gather}\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\r\n\\overrightarrow{\\text{OQ}} = \\dfrac{x+a}{x} \\overrightarrow{b}\r\n\\]\r\n\u4ee5\u4e0a\u3088\u308a\r\n\\[\\begin{align}\r\n\\left| \\overrightarrow{\\text{OP}} \\right| & = x+a , \\\\\r\n\\left| \\overrightarrow{\\text{OQ}} \\right| & = \\dfrac{b (x+a)}{x} , \\\\\r\n\\overrightarrow{\\text{OP}} \\cdot \\overrightarrow{\\text{OQ}} & = \\left| \\overrightarrow{\\text{OP}} \\right| \\left| \\overrightarrow{\\text{OQ}} \\right| \\cos \\theta \\\\\r\n& = \\dfrac{b (x+a)^2 \\cos \\theta}{x}\r\n\\end{align}\\]\r\n\u306a\u306e\u3067\r\n\\[\\begin{align}\r\nf(x) & = \\left| \\overrightarrow{\\text{PQ}} \\right|^2 \\\\\r\n& = \\left| \\overrightarrow{\\text{OP}} \\right|^2 +\\left| \\overrightarrow{\\text{OQ}} \\right|^2 -2 \\overrightarrow{\\text{OP}} \\cdot \\overrightarrow{\\text{OQ}} \\\\\r\n& = (x+a)^2 +\\dfrac{a^2 (x+a)^2}{x^2} -\\dfrac{2b (x+a)^2 \\cos \\theta}{x} \\\\\r\n& = \\underline{(x+a)^2 \\left( \\dfrac{b^2}{x^2} -\\dfrac{2b \\cos \\theta}{x} +1 \\right)}\r\n\\end{align}\\]\r\n

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

    \\(c = \\cos \\theta\\) \u3068\u304a\u304f\u3068, \\(0 \\lt \\theta \\leqq \\dfrac{\\pi}{2}\\) \u306a\u306e\u3067 \\(0 \\leqq c \\lt 1\\) ... [1] .
    \r\n(1)<\/strong> \u306e\u7d50\u679c\u3088\u308a\r\n\\[\\begin{align}\r\nf(x) & = \\left( x^2 +2ax +a^2 \\right) \\left( \\dfrac{b^2}{x^2} -\\dfrac{2bc}{x} +1 \\right) \\\\\r\n& = b^2 -2bcx +x^2 +\\dfrac{2ab^2}{x} -4abc +2ax +\\dfrac{a^2b^2}{x^2} -\\dfrac{2a^2bc}{x} +a^2 \\\\\r\n& = x^2 +2(a-bc)x +a^2+b^2+4abc +\\dfrac{2ab(b-ac)}{x} +\\dfrac{a^2b^2}{x^2}\r\n\\end{align}\\]\r\n\u306a\u306e\u3067\r\n\\[\\begin{align}\r\nf'(x) & = 2x +2(a-bc) -\\dfrac{2ab(b-ac)}{x^2} -\\dfrac{2a^2b^2}{x^3} , \\\\\r\nf''(x) & = 2 +\\dfrac{4ab(b-ac)}{x^3} +\\dfrac{6a^2b^2}{x^4}\r\n\\end{align}\\]\r\n\u3053\u3053\u3067, \\(a \\lt b\\) \u3068 [1] \u3088\u308a\r\n\\[\r\nb-ac \\gt b-a \\gt 0\r\n\\]\r\n\u306a\u306e\u3067, \\(f''(x)\\) \u306e\u3059\u3079\u3066\u306e\u9805\u306f\u6b63\u3060\u304b\u3089\r\n\\[\r\nf''(x) \\gt 0\r\n\\]\r\n

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

    \\(a=1\\) , \\(b=\\sqrt{6}\\) , \\(\\cos \\theta =\\dfrac{1}{\\sqrt{6}}\\) \u306e\u3068\u304d\r\n\\[\\begin{align}\r\nf(x) & = (x+1)^2 \\left( \\dfrac{6}{x^2} -\\dfrac{2}{x} +1 \\right) , \\\\\r\nf'(x) & = 2x +2 \\left( 1 -\\sqrt{6} \\cdot \\dfrac{1}{\\sqrt{6}}\\right) -\\dfrac{2\\sqrt{6} \\left( \\sqrt{6} -\\frac{1}{\\sqrt{6}} \\right)}{x^2} - \\dfrac{12}{x^3} \\\\\r\n& = 2x -\\dfrac{10}{x^2} -\\dfrac{12}{x^3} \\\\\r\n& = \\dfrac{2 ( x^4 -5x -6 )}{x^3} \\\\\r\n& = \\dfrac{2 (x+1)(x-2)( x^2+3x+3 )}{x^3}\r\n\\end{align}\\]\r\n\\(x \\gt 0\\) \u306b\u304a\u3044\u3066, \\(x^2+3x+3 \\gt 0\\) \u306a\u306e\u3067, \\(f(x)\\) \u306e\u5897\u6e1b\u8868\u306f\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b.\r\n\\[\r\n\\begin{array}{c|cccc} x & (0) & \\cdots & 2 & \\cdots \\\\ \\hline f'(x) & & - & 0 & + \\\\ \\hline f(x) & & - & \\text{\u6700\u5c0f} & + \\end{array}\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(f(x)\\) \u306e\u6700\u5c0f\u5024\u306f\r\n\\[\r\nf(2) = 3^2 \\left( \\dfrac{6}{2^2} -\\dfrac{2}{2} +1 \\right) = \\dfrac{27}{2}\r\n\\]\r\n\u3053\u306e\u3068\u304d PQ \u306e\u9577\u3055\u3082\u6700\u5c0f\u3068\u306a\u308b\u306e\u3067, \u6c42\u3081\u308b\u6700\u5c0f\u5024\u306f\r\n\\[\r\n\\sqrt{\\dfrac{27}{2}} = \\underline{\\dfrac{3 \\sqrt{6}}{2}}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u5e73\u9762\u4e0a\u306b \\(3\\) \u70b9 O , A , B \u304c\u3042\u308a, \\(\\text{OA} = a , \\ \\text{OB} = b \\ ( 0 \\lt a \\lt b )\\) \u3067, \\(\\overrightarrow{\\text … \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-243","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\/243","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=243"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/243\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=243"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=243"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=243"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}