{"id":1182,"date":"2015-07-21T23:57:06","date_gmt":"2015-07-21T14:57:06","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=1182"},"modified":"2021-09-29T23:38:38","modified_gmt":"2021-09-29T14:38:38","slug":"tbr201406","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tbr201406\/","title":{"rendered":"\u7b51\u6ce2\u5927\u7406\u7cfb2014\uff1a\u7b2c6\u554f"},"content":{"rendered":"
\n

\\(xy\\) \u5e73\u9762\u4e0a\u306b\u6955\u5186\r\n\\[\r\nC _ 1 : \\ \\dfrac{x^2}{a^2} +\\dfrac{y^2}{9} = 1 \\quad ( a \\gt \\sqrt{13} )\r\n\\]\r\n\u304a\u3088\u3073\u53cc\u66f2\u7dda\r\n\\[\r\nC _ 2 : \\ \\dfrac{x^2}{4} -\\dfrac{y^2}{b^2} = 1 \\quad \\left( b \\gt 0 \\right)\r\n\\]\r\n\u304c\u3042\u308a, \\(C _ 1\\) \u3068 \\(C _ 2\\) \u306f\u540c\u4e00\u306e\u7126\u70b9\u3092\u3082\u3064\u3068\u3059\u308b.\r\n\u307e\u305f \\(C _ 1\\) \u3068 \\(C _ 2\\) \u306e\u4ea4\u70b9 \\(P \\ \\left( 2 \\sqrt{1 +\\dfrac{t^2}{b^2}} , t \\right) \\ ( t \\gt 0 )\\) \u306b\u304a\u3051\u308b \\(C _ 1 , C _ 2\\) \u306e\u63a5\u7dda\u3092\u305d\u308c\u305e\u308c \\(\\ell _ 1 , \\ell _ 2\\) \u3068\u3059\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(a\\) \u3068 \\(b\\) \u306e\u9593\u306b\u6210\u308a\u7acb\u3064\u95a2\u4fc2\u5f0f\u3092\u6c42\u3081, \u70b9 \\(P\\) \u306e\u5ea7\u6a19\u3092 \\(a\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(\\ell _ 1\\) \u3068 \\(\\ell _ 2\\) \u304c\u76f4\u4ea4\u3059\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\\(a\\) \u304c \\(a \\gt \\sqrt{13}\\) \u3092\u6e80\u305f\u3057\u306a\u304c\u3089\u52d5\u304f\u3068\u304d\u306e\u70b9 \\(P\\) \u306e\u8ecc\u8de1\u3092\u56f3\u793a\u305b\u3088.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

    \u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n

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

    \\(3 \\lt \\sqrt{13} \\lt a\\) \u306a\u306e\u3067, \\(C _ 1\\) \u306e\u7126\u70b9\u306f\r\n\\[\r\n\\left( \\pm \\sqrt{a^2 -9} , 0 \\right)\r\n\\]\r\n\\(C _ 2\\) \u306e\u7126\u70b9\u306f\r\n\\[\r\n\\left( \\pm \\sqrt{4 +b^2} , 0 \\right)\r\n\\]\r\n\u3053\u306e \\(2\\) \u70b9\u304c\u4e00\u81f4\u3059\u308b\u306e\u3067\r\n\\[\\begin{align}\r\na^2 -9 & = 4 +b^2 \\\\\r\n\\text{\u2234} \\quad b & = \\underline{\\sqrt{a^2 -13}} \\quad ( \\ \\text{\u2235} \\ b \\gt 0 \\ ) ... [1]\r\n\\end{align}\\]\r\n\\(C _ 1 , C _ 2\\) \u306e\u5f0f\u304b\u3089, \\(x\\) \u3092\u6d88\u53bb\u3059\u308b\u3068\r\n\\[\\begin{align}\r\na^2 \\left( 1 -\\dfrac{y^2}{9} \\right) & = 4 \\left( 1 +\\dfrac{y^2}{b^2} \\right) \\\\\r\na^2 b^2 ( 9 -y^2 ) & = 9 b^2 ( b^2 +y^2 ) \\\\\r\n( 36 +a^2 b^2 ) y^2 & = 9 b^2 ( a^2 -4 )\r\n\\end{align}\\]\r\n\u3053\u308c\u306b [1] \u3092\u4ee3\u5165\u3059\u308b\u3068\r\n\\[\\begin{align}\r\n\\{ 36 +a^2 ( a^2 -13 ) \\} y^2 & = 9 ( a^2 -13 ) ( a^2 -4 ) \\\\\r\n( a^2-4 ) ( a^2 -9 ) y^2 & = 9 ( a^2 -13 ) ( a^2 -4 ) \\\\\r\n\\text{\u2234} \\quad y^2 & = \\dfrac{9 ( a^2 -13 )}{a^2 -9} \\quad ( \\ \\text{\u2235} \\ a^2 \\neq 4 \\ )\r\n\\end{align}\\]\r\n\\(y \\gt 0\\) \u306a\u306e\u3067\r\n\\[\r\ny = \\dfrac{3 \\sqrt{a^2 -13}}{\\sqrt{a^2 -9}}\r\n\\]\r\n\u3053\u306e\u3068\u304d\r\n\\[\\begin{align}\r\nx & = 2 \\sqrt{1 +\\dfrac{y^2}{a^2 -13}} \\\\\r\n& = 2 \\sqrt{1 +\\dfrac{9}{a^2 -9}} \\\\\r\n& = \\dfrac{2a}{\\sqrt{a^2 -9}}\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, P \u306e\u5ea7\u6a19\u306f\r\n\\[\r\n\\underline{\\left( \\dfrac{2a}{\\sqrt{a^2 -9}} , \\dfrac{3 \\sqrt{a^2 -13}}{\\sqrt{a^2 -9}} \\right)}\r\n\\]\r\n

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

    \\(P \\ ( s , t )\\) \u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\n\\ell _ 1 : \\ & \\dfrac{sx}{a^2} +\\dfrac{ty}{9} = 1 , \\\\\r\n\\ell _ 2 : \\ & \\dfrac{sx}{4} -\\dfrac{ty}{b^2} = 1\r\n\\end{align}\\]\r\n\\(\\ell _ 1 , \\ell _ 2\\) \u306e\u6cd5\u7dda\u30d9\u30af\u30c8\u30eb\u3092\u305d\u308c\u305e\u308c \\(\\overrightarrow{h _ 1} , \\overrightarrow{h _ 2}\\) \u3068\u304a\u3051\u3070\r\n\\[\r\n\\overrightarrow{h _ 1} = \\left( \\dfrac{s}{a^2} , \\dfrac{t}{9} \\right) , \\ \\overrightarrow{h _ 2} = \\left( \\dfrac{s}{4} , -\\dfrac{t}{b^2} \\right)\r\n\\]\r\n\u306a\u306e\u3067\r\n\\[\\begin{align}\r\n\\overrightarrow{h _ 1} \\cdot \\overrightarrow{h _ 2} & = \\dfrac{s^2}{4 a^2} -\\dfrac{t^2}{9 b^2} \\\\\r\n& = \\dfrac{1}{4 a^2} \\cdot \\dfrac{4 a^2}{a^2 -9} -\\dfrac{1}{9 b^2} \\cdot \\dfrac{9 b^2}{a^2 -9} \\\\\r\n& = 0\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, \\(\\overrightarrow{h _ 1} \\perp \\overrightarrow{h _ 2}\\) \u306a\u306e\u3067, \\(\\ell _ 1 \\perp \\ell _ 2\\) .<\/p>\r\n

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

    \\[\\begin{align}\r\ns^2 & = \\dfrac{4a^2}{a^2 -9} \\quad ... [2] , \\\\\r\nt^2 & = \\dfrac{9 ( a^2 -13 )}{a^2 -9} \\quad ... [3]\r\n\\end{align}\\]\r\n[2] \u3088\u308a\r\n\\[\\begin{align}\r\ns^2 a^2 -9 s^2 & = 4 a^2 \\\\\r\n\\text{\u2234} \\quad a^2 & = \\dfrac{9 s^2}{s^2 -4}\r\n\\end{align}\\]\r\n\u3053\u308c\u3092 [3] \u306b\u4ee3\u5165\u3059\u308c\u3070\r\n\\[\\begin{align}\r\nt^2 & = \\dfrac{9 \\left( \\frac{9 s^2}{s^2 -4} -13 \\right)}{\\frac{9 s^2}{s^2 -4} -9} \\\\\r\n& = \\dfrac{9 s^2 -13 ( s^2 -4 )}{4} \\\\\r\n& = 13 -s^2 \\\\\r\n\\text{\u2234} \\quad & s^2 +t^2 = 13\r\n\\end{align}\\]\r\n\u307e\u305f, [2] \u3088\u308a\r\n\\[\r\ns^2 = 4 +\\dfrac{36}{a^2 -9}\r\n\\]\r\n\u3067\u3042\u308a, \\(a \\gt \\sqrt{13}\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\n4 \\lt s^2 & \\lt 4 +\\dfrac{36}{13 -9} = 13 \\\\\r\n\\text{\u2234} \\quad 2 & \\lt s \\lt \\sqrt{13}\r\n\\end{align}\\]\r\n\u4ee5\u4e0a\u3088\u308a, \u70b9 \\(P\\) \u306e\u8ecc\u8de1\u306f\r\n\\[\r\n\\text{\u5186} : \\ x^2 +y^2 = 13 \\ \\left( 2 \\lt x \\lt \\sqrt{13} \\right)\r\n\\]\r\n\u3067\u3042\u308a, \u56f3\u793a\u3059\u308c\u3070\u4e0b\u56f3\u5b9f\u7dda\u90e8\uff08\u25cb\u306f\u542b\u307e\u306a\u3044\uff09.<\/p>\r\n\"tbr20140601\"\r\n","protected":false},"excerpt":{"rendered":"\\(xy\\) \u5e73\u9762\u4e0a\u306b\u6955\u5186 \\[ C _ 1 : \\ \\dfrac{x^2}{a^2} +\\dfrac{y^2}{9} = 1 \\quad ( a \\gt \\sqrt{13} ) \\] \u304a\u3088\u3073\u53cc\u66f2\u7dda \\[ C _ 2 : … \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":[121],"tags":[144,112],"class_list":["post-1182","post","type-post","status-publish","format-standard","hentry","category-tsukuba_r_2014","tag-tsukuba_r","tag-112"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1182","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=1182"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1182\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1182"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1182"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1182"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}