{"id":130,"date":"2011-11-30T01:05:31","date_gmt":"2011-11-29T16:05:31","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=130"},"modified":"2021-09-24T18:06:07","modified_gmt":"2021-09-24T09:06:07","slug":"tok201004","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tok201004\/","title":{"rendered":"\u6771\u5de5\u59272010\uff1a\u7b2c4\u554f"},"content":{"rendered":"<hr \/>\n<p>\\(a\\) \u3092\u6b63\u306e\u5b9a\u6570\u3068\u3059\u308b.\r\n\u539f\u70b9\u3092 \\(O\\) \u3068\u3059\u308b\u5ea7\u6a19\u5e73\u9762\u4e0a\u306b\u5b9a\u70b9 \\(A = A \\ ( a, 0 )\\) \u3068, \\(A\\) \u3068\u7570\u306a\u308b\u52d5\u70b9 \\(P = P \\ ( x, y )\\) \u3092\u3068\u308b.\r\n\u6b21\u306e\u6761\u4ef6<br \/>\r\n\u3000\u3000\\(A\\) \u304b\u3089 \\(P\\) \u306b\u5411\u3051\u305f\u534a\u76f4\u7dda\u4e0a\u306e\u70b9 \\(Q\\) \u306b\u5bfe\u3057\r\n\\[\r\n\\dfrac{AQ}{AP} \\leqq 2 \\ \\text{\u306a\u3089\u3070} \\ \\dfrac{QP}{OQ} \\leqq \\dfrac{AP}{OA}\r\n\\]\r\n\u3092\u6e80\u305f\u3059 \\(P\\) \u304b\u3089\u306a\u308b\u9818\u57df\u3092 \\(D\\) \u3068\u3059\u308b. \\(D\\) \u3092\u56f3\u793a\u305b\u3088.<\/p>\r\n<hr>\r\n<!--more-->\r\n<p><strong>\u3010 \u89e3 \u7b54 \u3011<\/strong><\/p>\r\n<p>\\(\\overrightarrow{AQ} = t \\overrightarrow{AP}\\) \u3068\u304a\u304f\u3068, \\(\\dfrac{AQ}{AP} \\leqq 2\\) \u3088\u308a\r\n\\[\r\n0 \\leqq t \\leqq 2\r\n\\]\r\n\u6761\u4ef6\u3088\u308a, \\(\\dfrac{AQ}{AP} \\leqq 2\\) \u306e\u3068\u304d\u306b, \u70b9 \\(Q\\) \u304c\u70b9 \\(O\\) \u3068\u4e00\u81f4\u3057\u3066\u306f\u3044\u3051\u306a\u3044.<br \/>\r\n\u3053\u308c\u306f, \u70b9 \\(P\\) \u304c \\(x\\) \u8ef8\u4e0a\u306b\u3042\u3063\u3066\uff08 \\(y = 0\\) \uff09\r\n\\[\\begin{align}\r\na +2(x-a) & \\leqq 0 \\\\\r\n\\text{\u2234} \\quad x \\leqq \\dfrac{a}{2}\r\n\\end{align}\\]\r\n\u306e\u3068\u304d\u3067\u3042\u308b. \u3000... [1]\r\n\\(\\overrightarrow{OA} = \\overrightarrow{a}\\) , \\(\\overrightarrow{OP} = \\overrightarrow{p}\\) \u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\n\\overrightarrow{AP} & = \\overrightarrow{p} -\\overrightarrow{a} , \\quad \\overrightarrow{AQ} = t \\left( \\overrightarrow{p} -\\overrightarrow{a} \\right) , \\\\\r\n\\overrightarrow{OQ} & = \\overrightarrow{OA} +\\overrightarrow{AQ} = t \\overrightarrow{p} -( t-1 ) \\overrightarrow{a} , \\\\\r\n\\overrightarrow{PQ} & = \\overrightarrow{OQ} -\\overrightarrow{OP} = ( t-1 ) \\left( \\overrightarrow{p} -\\overrightarrow{a} \\right)\r\n\\end{align}\\]\r\n\u4ee5\u4e0a\u3092\u7528\u3044\u308b\u3068, \\(\\dfrac{QP}{OQ} \\leqq \\dfrac{AP}{OA}\\) \u3088\u308a\r\n\\[\\begin{align}\r\n\\left| \\overrightarrow{PQ} \\right|^2 \\cdot \\left| \\overrightarrow{OA} \\right|^2 \\leqq \\left| \\overrightarrow{AP} \\right|^2 & \\cdot \\left| \\overrightarrow{OQ} \\right|^2 \\\\\r\nt^2 \\left| \\overrightarrow{AP} \\right|^2 \\cdot \\left| \\overrightarrow{a} \\right|^2 \\leqq \\left| \\overrightarrow{AP} \\right|^2 & \\cdot \\left| t \\overrightarrow{p} -( t-1 ) \\overrightarrow{a} \\right|^2 \\\\\r\nt^2 \\left| \\overrightarrow{p} \\right|^2 -2t( t-1 ) \\overrightarrow{a} \\cdot \\overrightarrow{p} & \\geqq 0 \\\\\r\nt \\left\\{ ( x^2+y^2 ) t -2ax( t-1 ) \\right\\} & \\geqq 0 \\quad ... [ \\text{A} ]\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(0 \\leqq t \\leqq 2\\) \u306e\u3068\u304d, [A] \u3092\u6e80\u305f\u3059\u6b63\u306e\u5b9f\u6570 \\(a\\) \u304c\u5b58\u5728\u3059\u308b\u305f\u3081\u306e \\(x , y\\) \u306e\u6761\u4ef6\u3092\u6c42\u3081\u308c\u3070\u3088\u3044.<\/p>\r\n<ol>\r\n<li><p><strong>1*<\/strong>\u3000\\(t = 0\\) \u306e\u3068\u304d\r\n\\(x , y\\) \u306b\u3088\u3089\u305a [A] \u306f\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n<li><p><strong>2*<\/strong>\u3000\\(0 \\lt t \\leqq 2\\) \u306e\u3068\u304d<br \/>\r\n[A] \u3088\u308a\r\n\\[\r\n( x^2+y^2 ) t -2ax( t-1 ) \\geqq 0\r\n\\]\r\n\u5de6\u8fba\u3092 \\(t\\) \u306e \\(1\\) \u6b21\u95a2\u6570\u3068\u307f\u306a\u3057\u3066 \\(f(t)\\) \u3068\u304a\u304f\u3068\r\n\\[\r\nf(t) = ( x^2-2ax +y^2 ) t +2ax\r\n\\]\r\n\u3053\u308c\u304c \\(0 \\lt t \\leqq 2\\) \u306b\u304a\u3044\u3066 \\(f(t) \\geqq 0\\) \u3068\u306a\u308b\u6761\u4ef6\u306f, \u300c \\(f(0) \\geqq 0\\) \u304b\u3064 \\(f(2) \\geqq 0\\) \u300d ... [2] .\r\n<ul>\r\n<li>\\(f(0) \\geqq 0\\) \u3088\u308a\r\n\\[\\begin{align}\r\n2ax & \\geqq 0 \\\\\r\n\\text{\u2234} \\quad x & \\geqq 0 \\quad ( \\ \\text{\u2235} \\ a \\gt 0 )\r\n\\end{align}\\]<\/li>\r\n<li>\\(f(2) \\geqq 0\\) \u3088\u308a\r\n\\[\\begin{align}\r\n2( x^2-ax +y^2 ) & \\geqq 0 \\\\\r\n\\left( x -\\dfrac{a}{2} \\right)^2 +y^2 & \\geqq \\left( \\dfrac{a}{2} \\right)^2\r\n\\end{align}\\]<\/li>\r\n<\/ul>\r\n\u3057\u305f\u304c\u3063\u3066, [2] \u306f\r\n\\[\r\nx \\geqq 0 \\ \\text{\u304b\u3064} \\ \\left( x -\\dfrac{a}{2} \\right)^2 +y^2 \\geqq \\left( \\dfrac{a}{2} \\right)^2\r\n\\]<\/li>\r\n<\/ol>\r\n[1] , <strong>1*<\/strong> , <strong>2*<\/strong> \u304a\u3088\u3073\u6761\u4ef6 \\(( x , y ) \\neq ( a , 0 )\\) \u304b\u3089,\r\n\u6c42\u3081\u308b\u9818\u57df\u306f\u4e0b\u56f3\u659c\u7dda\u90e8\uff08\u5883\u754c\u306f\u542b\u307f, \u70b9 \\(( 0 , 0 ) , ( a , 0 )\\) \u306f\u542b\u307e\u306a\u3044\uff09.<\/p>\r\n<img decoding=\"async\" src=\"\/\/www.roundown.net\/nyushi\/wp-content\/uploads\/tokodai2010_04_01.png\" alt=\"\" title=\"tokodai2010_04_01\" class=\"aligncenter size-full\" \/>\r\n","protected":false},"excerpt":{"rendered":"\\(a\\) \u3092\u6b63\u306e\u5b9a\u6570\u3068\u3059\u308b. \u539f\u70b9\u3092 \\(O\\) \u3068\u3059\u308b\u5ea7\u6a19\u5e73\u9762\u4e0a\u306b\u5b9a\u70b9 \\(A = A \\ ( a, 0 )\\) \u3068, \\(A\\) \u3068\u7570\u306a\u308b\u52d5\u70b9 \\(P = P \\ ( x, y )\\) \u3092\u3068\u308b. \u6b21\u306e\u6761\u4ef6 \u3000\u3000\\( &hellip; <a href=\"https:\/\/www.roundown.net\/nyushi\/tok201004\/\">\u7d9a\u304d\u3092\u8aad\u3080 <span class=\"meta-nav\">&rarr;<\/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":[25],"tags":[141,14],"class_list":["post-130","post","type-post","status-publish","format-standard","hentry","category-toko_2010","tag-toko","tag-14"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/130","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=130"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/130\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=130"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=130"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=130"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}