{"id":55,"date":"2011-11-26T18:20:25","date_gmt":"2011-11-26T09:20:25","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=55"},"modified":"2021-09-16T05:38:47","modified_gmt":"2021-09-15T20:38:47","slug":"ngr201103","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/ngr201103\/","title":{"rendered":"\u540d\u53e4\u5c4b\u5927\u7406\u7cfb2011\uff1a\u7b2c3\u554f"},"content":{"rendered":"<hr \/>\n<p>\\(xy\\) \u5e73\u9762\u4e0a\u306b \\(3\\) \u70b9 O \\(( 0 , 0 )\\) , A \\(( 1 , 0 )\\) , B \\(( 0 , 1 )\\) \u304c\u3042\u308b.<\/p>\r\n<ol>\r\n<li><p><strong>(1)<\/strong>\u3000\\(a \\gt 0\\) \u3068\u3059\u308b. \\(\\text{OP} : \\text{AP} = 1 : a\\) \u3092\u6e80\u305f\u3059\u70b9P\u306e\u8ecc\u8de1\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<li><p><strong>(2)<\/strong>\u3000\\(a \\gt 0 , \\ b \\gt 0\\) \u3068\u3059\u308b. \\(\\text{OP} : \\text{AP} : \\text{BP} = 1 : a : b\\) \u3092\u6e80\u305f\u3059\u70b9 P \u304c\u5b58\u5728\u3059\u308b\u305f\u3081\u306e \\(a , b\\) \u306b\u5bfe\u3059\u308b\u6761\u4ef6\u3092\u6c42\u3081, \\(ab\\) \u5e73\u9762\u4e0a\u306b\u56f3\u793a\u305b\u3088.<\/p><\/li>\r\n<\/ol>\r\n<hr \/>\r\n\r\n    <div id='extLink55'>\r\n        <p>\r\n            <a href='https:\/\/www.roundown.net\/nyushi\/ngr201103\/' name='ext55' onclick=\"showHide(55,'https:\/\/www.roundown.net\/nyushi\/ngr201103\/',this,'entry');return false;\">\u89e3\u7b54\u306f\u3053\u3061\u3089 &#187;<\/a>\r\n        <\/p>\r\n    <\/div>\r\n    <div id='extText55' style='display: none'>\r\n        \r\n<h2>\u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n<p><strong>(1)<\/strong><\/p>\r\n<p>P \\(( X , Y )\\) \u3068\u304a\u3051\u3070\r\n\\[\r\n\\text{OP} = \\sqrt{X^2+Y^2} , \\ \\text{AP} = \\sqrt{(X-1)^2+Y^2}\n\\]\r\n\\(\\text{OP} : \\text{AP} = 1 : a\\) \u3088\u308a, \\(\\text{AP}^2 = a^2 \\text{OP}^2\\) \u306a\u306e\u3067\r\n\\[\\begin{gather}\r\n(X-1)^2 +Y^2 = a^2 \\left( X^2+Y^2 \\right) \\\\\r\n(1-a^2)X^2 -2X +(1-a^2)Y^2+1 = 0\n\\end{gather}\\]\r\n<ol>\r\n<li><p><strong>1*<\/strong>\u3000\\(a = 1\\) \u306e\u3068\u304d\r\n\\[\\begin{align}\r\n-2X+1& = 0 \\\\\r\n\\text{\u2234} \\quad X & = \\dfrac{1}{2}\n\\end{align}\\]<\/li>\r\n<li><p><strong>2*<\/strong>\u3000\\(a \\neq 1\\) \u306e\u3068\u304d\r\n\\[\\begin{align}\r\nX^2 -\\dfrac{2X}{1-a^2} +Y^2 +\\dfrac{1}{1-a^2} & = 0 \\\\\r\n\\text{\u2234} \\quad \\left( X -\\dfrac{1}{1-a^2} \\right)^2 +Y^2 & = \\dfrac{a^2}{1-a^2}\n\\end{align}\\]<\/li>\r\n<\/ol>\r\n<p><strong>1*<\/strong> <strong>2*<\/strong> \u3088\u308a, \u70b9 P \u306e\u8ecc\u8de1\u306f\r\n\\[\r\n\\underline{\\left\\{ \\begin{array}{ll} \\text{\u76f4\u7dda} : \\ x = \\dfrac{1}{2} \\quad ... [1] & ( \\ a = 1 \\text{\u306e\u3068\u304d} ) \\\\ \\text{\u5186} : \\ \\left( x -\\dfrac{1}{1-a^2} \\right)^2 +y^2 = \\dfrac{a^2}{1-a^2} \\quad ... [2] & ( \\ a \\neq 1 \\text{\u306e\u3068\u304d} ) \\end{array} \\right.}\n\\]\r\n<p><strong>(2)<\/strong><\/p>\r\n<p>\\(\\text{OQ} : \\text{BQ} = 1 : b\\) \u3092\u6e80\u305f\u3059\u70b9 Q \u306e\u8ecc\u8de1\u306f, <strong>(1)<\/strong> \u3068\u540c\u69d8\u306b\u8003\u3048\u308c\u3070\r\n\\[\r\n\\underline{\\left\\{ \\begin{array}{ll} \\text{\u76f4\u7dda} : \\ y = \\dfrac{1}{2} \\quad ... [3] & ( \\ b = 1 \\text{\u306e\u3068\u304d} ) \\\\ \\text{\u5186} : \\ x^2 +\\left( y -\\dfrac{1}{1-b^2} \\right)^2 = \\dfrac{b^2}{1-b^2} \\quad ... [4] & ( \\ b \\neq 1 \\text{\u306e\u3068\u304d} ) \\end{array} \\right.}\n\\]\r\nP \u3068 Q \u306e\u8ecc\u8de1\u304c, \u5171\u6709\u70b9\u3092\u6301\u3064\u305f\u3081\u306e \\(a , b\\) \u306e\u6761\u4ef6\u3092\u6c42\u3081\u308c\u3070\u3088\u3044.<\/p>\r\n<ol>\r\n<li><p><strong>1*<\/strong>\u3000\\(a = b = 1\\) \u306e\u3068\u304d<br \/>\r\nP , Q \u306e\u8ecc\u8de1, [1] \u3068 [3] \u306f, \u5171\u6709\u70b9 \\(\\left( \\dfrac{1}{2} , \\dfrac{1}{2} \\right)\\) \u3092\u3082\u3064.<\/p><\/li>\r\n<li><p><strong>2*<\/strong>\u3000\\(a \\neq 1 , \\ b = 1\\) \u306e\u3068\u304d<br \/>\r\nP , Q \u306e\u8ecc\u8de1, [2] \u3068 [3] \u304c\u5171\u6709\u70b9\u3092\u3082\u3064\u6761\u4ef6\u306f\r\n\\[\r\n( \\text{\u5186 [2] \u306e\u4e2d\u5fc3\u3068\u76f4\u7dda [3] \u306e\u8ddd\u96e2} ) \\leqq ( \\text{\u5186 [2] \u306e\u534a\u5f84} )\r\n\\]\r\n\u306a\u306e\u3067\r\n\\[\\begin{align}\r\n\\dfrac{1}{2} & \\leqq \\dfrac{a}{\\left| 1-a^2 \\right|} \\\\\r\n\\left| 1-a^2 \\right| & \\leqq 2a \\\\\r\n-2a \\leqq 1-a^2 & \\leqq 2a \\\\\r\n\\text{\u300c} a^2-2a-1 \\leqq 0 & \\text{\u300d\u304b\u3064\u300c} a^2+2a-1 \\geqq 0 \\text{\u300d} \\\\\r\n\\text{\u300c} 1 -\\sqrt{2} \\leqq a \\leqq 1+\\sqrt{2} & \\text{\u300d\u304b\u3064\u300c} a \\leqq -1-\\sqrt{2} , -1+\\sqrt{2} \\leqq a \\text{\u300d} \\\\\r\n\\text{\u2234} \\quad \\sqrt{2}-1 & \\leqq a \\leqq \\sqrt{2}+1\n\\end{align}\\]<\/li>\r\n<li><p><strong>3*<\/strong>\u3000\\(a = 1 , \\ b \\neq 1\\) \u306e\u3068\u304d<br \/>\r\nP , Q \u306e\u8ecc\u8de1, [1] \u3068 [4] \u304c\u5171\u6709\u70b9\u3092\u3082\u3064\u6761\u4ef6\u306f, <strong>2*<\/strong> \u306e\u3068\u304d\u3068\u540c\u69d8\u306b\u3057\u3066\r\n\\[\r\n\\sqrt{2}-1 \\leqq b \\leqq \\sqrt{2}+1\n\\]<\/li>\r\n<li><p><strong>4*<\/strong>\u3000\\(a \\neq 1 , \\ b \\neq 1\\) \u306e\u3068\u304d<br \/>\r\nP , Q \u306e\u8ecc\u8de1, [2] \u3068 [4] \u304c\u5171\u6709\u70b9\u3092\u3082\u3064\u6761\u4ef6\u306f\r\n\\[\r\n( \\text{\u5186 [2] [4] \u306e\u534a\u5f84\u306e\u5dee} ) \\leqq ( \\text{\u5186 [2] [4] \u306e\u4e2d\u5fc3\u9593\u306e\u8ddd\u96e2} ) \\leqq ( \\text{\u5186 [2] [4] \u306e\u534a\u5f84\u306e\u548c} )\r\n\\]\r\n\u306a\u306e\u3067, \u5404\u8fba\u3092\u5e73\u65b9\u3057\u3066\r\n\\[\\begin{align}\r\n\\left( \\dfrac{a}{\\left| 1-a^2 \\right|} -\\dfrac{b}{\\left| 1-b^2 \\right|} \\right)^2 & \\leqq \\dfrac{1}{(1-a^2)^2} +\\dfrac{1}{(1-b^2)^2} \\leqq \\left( \\dfrac{a}{\\left| 1-a^2 \\right|} +\\dfrac{b}{\\left| 1-b^2 \\right|} \\right)^2 \\\\\r\n-\\dfrac{2ab}{\\left| 1-a^2 \\right| \\left| 1-b^2 \\right|} & \\leqq \\dfrac{1}{(1-a^2)^2} +\\dfrac{1}{(1-b^2)^2} \\\\\r\n& \\qquad -\\left\\{ \\dfrac{a^2}{(1-a^2)^2} +\\dfrac{b^2}{(1-b^2)^2} \\right\\} \\leqq \\dfrac{2ab}{\\left| 1-a^2 \\right| \\left| 1-b^2 \\right|} \\\\\r\n-\\dfrac{2ab}{\\left| 1-a^2 \\right| \\left| 1-b^2 \\right|} & \\leqq \\dfrac{1}{1-a^2} +\\dfrac{1}{1-b^2} \\leqq \\dfrac{2ab}{\\left| 1-a^2 \\right| \\left| 1-b^2 \\right|}\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\r\n\\left| \\dfrac{1}{1-a^2} +\\dfrac{1}{1-b^2} \\right| \\leqq \\dfrac{2ab}{\\left| 1-a^2 \\right| \\left| 1-b^2 \\right|}\n\\]\r\n\\(\\left| 1-a^2 \\right| \\left| 1-b^2 \\right| \\gt 0\\) \u306a\u306e\u3067, \u5404\u8fba\u306e\u5206\u6bcd\u3092\u6255\u3063\u3066\r\n\\[\\begin{align}\r\n\\left| (1-b^2) +(1-a^2) \\right| & \\leqq 2ab \\\\\r\n-2ab \\leqq a^2+b^2-2 & \\leqq 2ab \\\\\r\n\\text{\u300c} (a+b)^2-2 \\geqq 0 \\text{\u300d\u304b\u3064\u300c} & (a-b)^2-2 \\leqq 0 \\text{\u300d} \\\\\r\n\\text{\u300c} a+b-\\sqrt{2} \\geqq 0 \\text{\u300d\u304b\u3064\u300c} & (a-b+\\sqrt{2})(a-b-\\sqrt{2}) \\leqq 0 \\text{\u300d} \\quad ( \\ \\text{\u2235} \\ a+b+\\sqrt{2} \\gt 0 ) \\\\\r\n\\text{\u2234} \\quad a+b \\geqq \\sqrt{2} , \\ a-\\sqrt{2} & \\leqq b \\leqq a+\\sqrt{2}\n\\end{align}\\]<\/li>\r\n<\/ol>\r\n<p><strong>1*<\/strong> \uff5e <strong>4*<\/strong> \u3088\u308a, \\(a , b\\) \u306e\u6761\u4ef6\u306f\u4e0b\u56f3\u659c\u7dda\u90e8\uff08\u5883\u754c\u90e8\u306f\u542b\u307f, \u767d\u70b9\u306f\u542b\u307e\u306a\u3044\uff09.<\/p>\r\n<img decoding=\"async\" src=\"\/\/www.roundown.net\/nyushi\/wp-content\/uploads\/nagoya_r_201103_01.png\" alt=\"\" title=\"nagoya_r_201103_01\" class=\"aligncenter size-full\" \/>\r\n\r\n        <p>\r\n            <a href='#ext55' onclick=\"showHide(55,0,this,'entry');return true;\">&#171; \u89e3\u7b54\u3092\u96a0\u3059 <\/a>\r\n        <\/p>\r\n    <\/div>","protected":false},"excerpt":{"rendered":"\\(xy\\) \u5e73\u9762\u4e0a\u306b \\(3\\) \u70b9 O \\(( 0 , 0 )\\) , A \\(( 1 , 0 )\\) , B \\(( 0 , 1 )\\) \u304c\u3042\u308b. (1)\u3000\\(a \\gt 0\\) \u3068\u3059\u308b. \\(\\text{OP}  [&hellip;]","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":[35],"tags":[143,13],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/55"}],"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=55"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/55\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=55"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=55"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=55"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}