{"id":1007,"date":"2015-04-04T22:49:27","date_gmt":"2015-04-04T13:49:27","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=1007"},"modified":"2021-03-09T16:26:56","modified_gmt":"2021-03-09T07:26:56","slug":"tkr201401","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tkr201401\/","title":{"rendered":"\u6771\u5927\u7406\u7cfb2014\uff1a\u7b2c1\u554f"},"content":{"rendered":"
\n

\\(1\\) \u8fba\u306e\u9577\u3055\u304c \\(1\\) \u306e\u6b63\u65b9\u5f62\u3092\u5e95\u9762\u3068\u3059\u308b\u56db\u89d2\u67f1 OABC-DEFG \u3092\u8003\u3048\u308b. \r\n\\(3\\) \u70b9 P , Q , R \u3092, \u305d\u308c\u305e\u308c\u8fba AE , \u8fba BF , \u8fba CG \u4e0a\u306b, \\(4\\) \u70b9 O , P , Q , R \u304c\u540c\u4e00\u5e73\u9762\u4e0a\u306b\u3042\u308b\u3088\u3046\u306b\u3068\u308b. \r\n\u56db\u89d2\u5f62 OPQR \u306e\u9762\u7a4d\u3092 \\(S\\) \u3068\u304a\u304f. \u307e\u305f, \\(\\angle \\text{AOP}\\) \u3092 \\(\\alpha\\) , \\(\\angle \\text{COR}\\) \u3092 \\(\\beta\\) \u3068\u304a\u304f.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(S\\) \u3092 \\(\\tan \\alpha\\) \u3068 \\(\\tan \\beta\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(\\alpha +\\beta = \\dfrac{\\pi}{4}\\) , \\(S = \\dfrac{7}{6}\\) \u3067\u3042\u308b\u3068\u304d, \\(\\tan \\alpha +\\tan \\beta\\) \u306e\u5024\u3092\u6c42\u3081\u3088. \u3055\u3089\u306b, \\(\\alpha \\leqq \\beta\\) \u306e\u3068\u304d, \\(\\tan \\alpha\\) \u306e\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

    OA , OC , OD \u304c\u305d\u308c\u305e\u308c \\(x\\) \u8ef8, \\(y\\) \u8ef8, \\(z\\) \u8ef8\u3068\u306a\u308b\u3088\u3046\u306b\u7a7a\u9593\u5ea7\u6a19\u3092\u3068\u308b.
    \r\n\u6761\u4ef6\u3088\u308a\r\n\\[\r\n\\overrightarrow{\\text{OP}} = \\left( \\begin{array}{c} 1 \\\\ 0 \\\\ \\tan \\alpha \\end{array} \\right) , \\quad \\overrightarrow{\\text{OQ}} = \\left( \\begin{array}{c} 0 \\\\ 1 \\\\ \\tan \\beta \\end{array} \\right)\r\n\\]\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u9762\u7a4d \\(S\\) \u306f\r\n\\[\\begin{align}\r\nS & = \\sqrt{\\left| \\overrightarrow{\\text{OP}} \\right|^2 \\left| \\overrightarrow{\\text{OQ}} \\right|^2 -\\left( \\overrightarrow{\\text{OP}} \\cdot \\overrightarrow{\\text{OQ}} \\right)^2} \\\\\r\n& = \\sqrt{\\left( 1 +\\tan^2 \\alpha \\right) \\left( 1 +\\tan^2 \\beta \\right) -\\left( \\tan \\alpha \\tan \\beta \\right)^2} \\\\\r\n& = \\underline{\\sqrt{1 +\\tan^2 \\alpha +\\tan^2 \\beta}}\r\n\\end{align}\\]\r\n

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

    \\(s = \\tan \\alpha +\\tan \\beta\\) , \\(t = \\tan \\alpha \\tan \\beta\\) \u3068\u304a\u304f.
    \r\n\u6761\u4ef6 \\(\\alpha +\\beta = \\dfrac{\\pi}{4}\\) \u3068\u52a0\u6cd5\u5b9a\u7406\u3088\u308a\r\n\\[\\begin{align}\r\n\\dfrac{s}{1-t} & = 1 \\\\\r\n\\text{\u2234} \\quad t & = 1-s \\quad ... [1]\r\n\\end{align}\\]\r\n(1)<\/strong> \u306e\u7d50\u679c\u3068\u6761\u4ef6 \\(S =\\dfrac{7}{6}\\) \u3088\u308a\r\n\\[\\begin{align}\r\n1 +s^2 -2t & = \\left( \\dfrac{7}{6} \\right)^2 \\\\\r\n36 \\left( s^2 +2s -1 \\right) & = 49 \\quad ( \\ \\text{\u2235} \\ [1] \\ ) \\\\\r\n36s^2 +72s -85 & = 0 \\\\\r\n(6s+17)(6s-5) & = 0 \\\\\r\n\\text{\u2234} \\quad s & = \\dfrac{5}{6} \\quad ( \\ \\text{\u2235} \\ s \\gt 0 \\ )\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\r\n\\tan \\alpha +\\tan \\beta = \\underline{\\dfrac{5}{6}}\r\n\\]\r\n[1] \u3088\u308a\r\n\\[\r\nt = 1 -\\dfrac{5}{6} = \\dfrac{1}{6}\r\n\\]\r\n\\(\\tan \\alpha\\) \u3068 \\(\\tan \\beta\\) \u306f, \\(X\\) \u306b\u3064\u3044\u3066\u306e \\(2\\) \u6b21\u65b9\u7a0b\u5f0f \\(X^2 -sX +t = 0\\) ... [A] \u306e\u89e3\u3067\u3042\u308b.
    \r\n[A] \u3092\u3068\u304f\u3068\r\n\\[\\begin{align}\r\n6 X^2 -5X +1 & = 0 \\\\\r\n(3X-1)(2X-1) & = 0 \\\\\r\n\\text{\u2234} \\quad X & = \\dfrac{1}{3} , \\dfrac{1}{2}\r\n\\end{align}\\]\r\n\\(0 \\lt \\alpha \\leqq \\beta \\lt \\dfrac{\\pi}{4}\\) \u3088\u308a, \\(\\tan \\alpha \\leqq \\tan \\beta\\) \u306a\u306e\u3067\r\n\\[\r\n\\tan \\alpha = \\underline{\\dfrac{1}{3}}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(1\\) \u8fba\u306e\u9577\u3055\u304c \\(1\\) \u306e\u6b63\u65b9\u5f62\u3092\u5e95\u9762\u3068\u3059\u308b\u56db\u89d2\u67f1 OABC-DEFG \u3092\u8003\u3048\u308b. \\(3\\) \u70b9 P , Q , R \u3092, \u305d\u308c\u305e\u308c\u8fba AE , \u8fba BF , \u8fba CG \u4e0a\u306b, \\(4\\) \u70b9 O , P … \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":[113],"tags":[139,112],"class_list":["post-1007","post","type-post","status-publish","format-standard","hentry","category-tokyo_r_2014","tag-tokyo_r","tag-112"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1007","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=1007"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1007\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1007"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1007"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1007"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}