{"id":199,"date":"2011-12-02T23:13:52","date_gmt":"2011-12-02T14:13:52","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=199"},"modified":"2021-09-16T06:23:46","modified_gmt":"2021-09-15T21:23:46","slug":"ngr200901","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/ngr200901\/","title":{"rendered":"\u540d\u53e4\u5c4b\u5927\u7406\u7cfb2009\uff1a\u7b2c1\u554f"},"content":{"rendered":"
\n

\\(a \\gt 0 , \\ b \\gt 0\\) \u3068\u3059\u308b. \u70b9 A \\((0, a)\\) \u3092\u4e2d\u5fc3\u3068\u3059\u308b\u534a\u5f84 \\(r\\) \u306e\u5186\u304c, \u53cc\u66f2\u7dda \\(x^2-\\dfrac{y^2}{b^2}=1\\) \u3068 \\(2\\) \u70b9 B \\((s, t)\\) , C \\((-s, t)\\) \u3067\u63a5\u3057\u3066\u3044\u308b. \u305f\u3060\u3057, \\(s \\gt 0\\) \u3068\u3059\u308b.\r\n\u3053\u3053\u3067, \u53cc\u66f2\u7dda\u3068\u5186\u304c\u70b9 P \u3067\u63a5\u3059\u308b\u3068\u306f, P \u304c\u53cc\u66f2\u7dda\u3068\u5186\u306e\u5171\u6709\u70b9\u3067\u3042\u308a, \u304b\u3064\u70b9 P \u306b\u304a\u3051\u308b\u53cc\u66f2\u7dda\u306e\u63a5\u7dda\u3068\u70b9 P \u306b\u304a\u3051\u308b\u5186\u306e\u63a5\u7dda\u304c\u4e00\u81f4\u3059\u308b\u3053\u3068\u3067\u3042\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(r , s , t\\) \u3092 \\(a\\) \u3068 \\(b\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\u25b3ABC \u304c\u6b63\u4e09\u89d2\u5f62\u3068\u306a\u308b \\(a\\) \u3068 \\(r\\) \u304c\u5b58\u5728\u3059\u308b\u3088\u3046\u306a \\(b\\) \u306e\u5024\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\"\"\r\n

    \\(C : \\ x^2 -\\dfrac{y^2}{b^2} = 1\\) , \\(D : \\ x^2+(y-a)^2 = r^2\\) \u3068\u304a\u304f.
    \r\n\\(C , D\\) \u306f \\(y\\) \u8ef8\u306b\u3064\u3044\u3066\u5bfe\u79f0\u306a\u306e\u3067, \\(x \\geqq 0\\) \u306b\u3064\u3044\u3066\u8003\u3048\u308c\u3070\u3088\u3044.
    \r\n\u70b9 \\(( s , t )\\) \u306b\u304a\u3051\u308b \\(C\\) \u306e\u63a5\u7dda\u306f\r\n\\[\r\nsx -\\dfrac{ty}{b^2} = 1 \\quad ... [1]\n\\]\r\n\\(D\\) \u306e\u63a5\u7dda\u306f\r\n\\[\\begin{align}\r\nsx +(t-a)(y-a) & =r^2 \\\\\r\n\\text{\u2234} \\quad sx +(t-a)y & = r^2+a(t-a) \\quad ... [2]\n\\end{align}\\]\r\n[1] \u3068 [2] \u304c\u4e00\u81f4\u3059\u308b\u306e\u3067,\r\n\\[\r\n\\left\\{ \\begin{array}{ll} t-a = -\\dfrac{t}{b^2} & ... [3] \\\\ r^2+a(t-a) =1 & ... [4] \\end{array} \\right.\n\\]\r\n[3] \u3088\u308a\r\n\\[\\begin{align}\r\n( b^2+1 )t & = ab^2 \\\\\r\n\\text{\u2234} \\quad t & = \\underline{\\dfrac{ab^2}{b^2+1}}\n\\end{align}\\]\r\n[4] \u306b\u4ee3\u5165\u3057\u3066\r\n\\[\\begin{align}\r\nr^2 & = 1-a \\cdot \\left( \\dfrac{a}{b^1+1} \\right) \\\\\r\n\\text{\u2234} \\quad r & =\\underline{\\sqrt{1 +\\dfrac{a^2}{b^2+1}}}\n\\end{align}\\]\r\n\\(C\\) \u306e\u5f0f\u3088\u308a\r\n\\[\r\ns =\\sqrt{1+ \\dfrac{t^2}{b^2}} =\\underline{\\sqrt{1+ \\left( \\dfrac{ab}{b^2+1} \\right)}}\n\\]\r\n

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

    \\(\\text{AB} = \\text{AC}\\) \u3067 BC \u306f \\(x\\) \u8ef8\u3068\u5e73\u884c\u306a\u306e\u3067, \u25b3ABC \u304c\u6b63\u4e09\u89d2\u5f62\u306b\u306a\u308b\u306e\u306f, AB \u306e\u50be\u304d\u304c \\(-\\sqrt{3}\\) \u306e\u3068\u304d.
    \r\nAB \u306f\u70b9 \\(( s, t )\\) \u306b\u304a\u3051\u308b \\(C\\) \u306e\u6cd5\u7dda\u3060\u304b\u3089, \u63a5\u7dda\u306e\u50be\u304d\u306f \\(\\dfrac{1}{\\sqrt{3}}\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\n\\dfrac{sb^2}{t} & = \\dfrac{b^2 \\sqrt{1+ \\left( \\dfrac{ab}{b^2+1} \\right)}}{\\dfrac{ab^2}{b^2+1}} \\\\\r\n& = \\dfrac{1}{a} \\sqrt{(b^2+1)^2 +a^2b^2} = \\dfrac{1}{\\sqrt{3}} \\\\\r\n\\text{\u2234} \\quad & 3(b^2+1)^2 +3a^2b^2 = a^2 \\\\\r\n\\text{\u2234} \\quad & a^2 = \\dfrac{3(b^2+1)^2}{1-3b^2} \\quad ... [5]\n\\end{align}\\]\r\n[5] \u306e\u5de6\u8fba, \u53f3\u8fba\u306e\u5206\u5b50\u306f\u6b63\u3067\u3042\u308b\u304b\u3089, [5] \u3092\u307f\u305f\u3059 \\(a\\) \u304c\u5b58\u5728\u3059\u308b\u6761\u4ef6\u306f\r\n\\[\\begin{align}\r\n& 1-3b^2 \\gt 0 \\\\\r\n\\text{\u2234} \\quad & \\underline{0 \\lt b \\lt \\dfrac{1}{\\sqrt{3}}} \\quad ( \\ \\text{\u2235} \\ b \\gt 0 )\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\\(a \\gt 0 , \\ b \\gt 0\\) \u3068\u3059\u308b. \u70b9 A \\((0, a)\\) \u3092\u4e2d\u5fc3\u3068\u3059\u308b\u534a\u5f84 \\(r\\) \u306e\u5186\u304c, \u53cc\u66f2\u7dda \\(x^2-\\dfrac{y^2}{b^2}=1\\) \u3068 \\(2\\) \u70b9 B \\(( … \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":[33],"tags":[143,15],"class_list":["post-199","post","type-post","status-publish","format-standard","hentry","category-nagoya_r_2009","tag-nagoya_r","tag-15"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/199","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=199"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/199\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=199"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=199"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=199"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}