{"id":7,"date":"2011-11-25T20:48:46","date_gmt":"2011-11-25T11:48:46","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=7"},"modified":"2021-03-12T16:53:08","modified_gmt":"2021-03-12T07:53:08","slug":"tkr201101","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tkr201101\/","title":{"rendered":"\u6771\u5927\u7406\u7cfb2011\uff1a\u7b2c1\u554f"},"content":{"rendered":"
\n

\u5ea7\u6a19\u5e73\u9762\u306b\u304a\u3044\u3066, \u70b9 P \\(( 0 , 1 )\\) \u3092\u4e2d\u5fc3\u3068\u3059\u308b\u534a\u5f84 \\(1\\) \u306e\u5186\u3092 \\(C\\) \u3068\u3059\u308b.\r\n\u3000\\(a\\) \u3092 \\(0 \\lt a \\lt 1\\) \u3092\u6e80\u305f\u3059\u5b9f\u6570\u3068\u3057, \u76f4\u7dda \\(y = a( x+1 )\\) \u3068 \\(C\\) \u3068\u306e\u4ea4\u70b9\u3092 Q , R \u3068\u3059\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(\\triangle \\text{PQR}\\) \u306e\u9762\u7a4d \\(S(a)\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(a\\) \u304c \\(0 \\lt a \\lt 1\\) \u306e\u7bc4\u56f2\u3092\u52d5\u304f\u3068\u304d, \\(S(a)\\) \u304c\u6700\u5927\u3068\u306a\u308b \\(a\\) \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\"todai_2011_01_01\"\r\n

    \u76f4\u7dda \\(l : \\ y = a( x+1 )\\) \u3068\u304a\u304f. \u70b9 P \u304b\u3089 \\(l\\) \u306b\u4e0b\u308d\u3057\u305f\u5782\u7dda\u306e\u8db3\u3092 H \u3068\u304a\u304f.
    \r\n\u70b9\u3068\u76f4\u7dda\u306e\u8ddd\u96e2\u306e\u516c\u5f0f\u3088\u308a\r\n\\[\r\n\\text{PH} = \\dfrac{\\left| a \\cdot 0 -1 \\cdot 1 +a \\right|}{\\sqrt{a^2 +1}} = \\dfrac{1-a}{\\sqrt{a^2 +1}} \\quad ( \\text{\u2235} \\ 0 \\lt a \\lt 1 )\r\n\\]\r\n\u25b3PQH \u304c\u76f4\u89d2\u4e09\u89d2\u5f62\u3067\u3042\u308b\u3053\u3068, \\(\\text{QH} = \\text{RH}\\) \u306b\u7740\u76ee\u3059\u308c\u3070\r\n\\[\r\n\\text{QR} = 2 \\sqrt{1 -\\left( \\dfrac{1-a}{\\sqrt{a^2 +1}} \\right)^2} = \\dfrac{2 \\sqrt{2a}}{\\sqrt{a^2 +1}}\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\r\nS(a) = \\dfrac{1}{2} \\text{QR} \\cdot \\text{PH} = \\underline{\\dfrac{\\sqrt{2a} ( 1-a )}{a^2 +1}}\r\n\\]\r\n

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

    \\[\r\nf(a) = \\dfrac{1}{2} \\left\\{ S(a) \\right\\}^2 = \\dfrac{a( 1-a )^2}{( a^2 +1 )^2}\r\n\\]\r\n\u3068\u304a\u304f\u3068, \\(f(a)\\) \u304c\u6700\u5927\u306e\u3068\u304d, \\(S(a)\\) \u3082\u6700\u5927\u3068\u306a\u308b.
    \r\n\u4ee5\u4e0b\u3067\u306f, \\(f(a)\\) \u306e\u6700\u5927\u3068\u306a\u308b\u3068\u304d\u306b\u3064\u3044\u3066\u8abf\u3079\u308b.\r\n\\[\\begin{align}\r\nf'(a) & = \\dfrac{\\left\\{ ( 1-a )^2 -2a( 1-a ) \\right\\}( a^2+1 )^2 -a( 1-a )^2 \\cdot 4a( a^2+1 )}{( a^2+1)^4} \\\\\r\n& = \\dfrac{( 1 -4a +3a^2 )( a^2+1 ) -4a^2 ( 1 -2a +a^2 )}{( a^2+1)^3} \\\\\r\n& = \\dfrac{( 3a^4 -4a^3+4a^2 -4a +1 ) -( 4a^4 -8a^3 +4a^2 )}{( a^2+1)^3} \\\\\r\n& = \\dfrac{-a^4 +4a^3 -4a +1}{( a^2+1)^3} \\\\\r\n& = \\dfrac{-( a^2-1 )( a^2+1 ) +4a( a^2+1 )}{( a^2+1)^3} \\\\\r\n& = \\dfrac{( 1-a^2 )( a^2-4a+1 )}{( a^2+1)^3} \\\\\r\n& = \\dfrac{( 1-a^2 )\\left\\{ a -\\left( 2+\\sqrt{3} \\right) \\right\\}\\left\\{ a -\\left( 2-\\sqrt{3} \\right) \\right\\}}{( a^2+1)^3}\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066 \\(0 \\lt a \\lt 1\\) \u306b\u304a\u3051\u308b \\(f(a)\\) \u306e\u5897\u6e1b\u306f\u4e0b\u8868\u306e\u3088\u3046\u306b\u306a\u308b.\r\n\\[\r\n\\begin{array}{c|ccccc} a & ( 0 ) & \\cdots & 2-\\sqrt{3} & \\cdots & ( 1 ) \\\\ \\hline f'(a) & & + & 0 & - & \\\\ \\hline f(a) & & \\nearrow & \\text{\u6700\u5927} & \\searrow & \\\\ \\end{array}\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066 \\(f(a)\\) \u306f \\(a = 2-\\sqrt{3}\\) \u306e\u3068\u304d\u6700\u5927\u3068\u306a\u308b.
    \r\n\u3086\u3048\u306b \\(S(a)\\) \u306f \\(a = \\underline{2-\\sqrt{3}}\\) \u306e\u3068\u304d\u6700\u5927\u3068\u306a\u308b.<\/p>\r\n","protected":false},"excerpt":{"rendered":"\u5ea7\u6a19\u5e73\u9762\u306b\u304a\u3044\u3066, \u70b9 P \\(( 0 , 1 )\\) \u3092\u4e2d\u5fc3\u3068\u3059\u308b\u534a\u5f84 \\(1\\) \u306e\u5186\u3092 \\(C\\) \u3068\u3059\u308b. \u3000\\(a\\) \u3092 \\(0 \\lt a \\lt 1\\) \u3092\u6e80\u305f\u3059\u5b9f\u6570\u3068\u3057, \u76f4\u7dda \\(y = a( x+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":[20],"tags":[139,13],"class_list":["post-7","post","type-post","status-publish","format-standard","hentry","category-tokyo_r_2011","tag-tokyo_r","tag-13"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/7","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=7"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/7\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=7"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=7"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=7"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}