{"id":82,"date":"2011-11-26T23:00:18","date_gmt":"2011-11-26T14:00:18","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=82"},"modified":"2021-09-30T10:02:56","modified_gmt":"2021-09-30T01:02:56","slug":"kbr201106","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/kbr201106\/","title":{"rendered":"\u7b51\u6ce2\u5927\u7406\u7cfb2011\uff1a\u7b2c6\u554f"},"content":{"rendered":"
\n

\\(d\\) \u3092\u6b63\u306e\u5b9a\u6570\u3068\u3059\u308b.\r\n\\(2\\) \u70b9 \\(A \\ ( -d , 0 )\\) , \\(B \\ ( d , 0 )\\) \u304b\u3089\u306e\u8ddd\u96e2\u306e\u548c\u304c \\(4d\\) \u3067\u3042\u308b\u70b9 \\(P\\) \u306e\u8ecc\u8de1\u3068\u3057\u3066\u5b9a\u307e\u308b\u6955\u5186 \\(E\\) \u3092\u8003\u3048\u308b. \u70b9 \\(A\\) , \u70b9 \\(B\\) , \u539f\u70b9 \\(O\\) \u304b\u3089\u6955\u5186 \\(E\\) \u4e0a\u306e\u70b9 \\(P\\) \u307e\u3067\u306e\u8ddd\u96e2\u3092\u305d\u308c\u305e\u308c \\(AP , BP , OP\\) \u3068\u66f8\u304f. \u3053\u306e\u3068\u304d, \u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\u6955\u5186 \\(E\\) \u306e\u9577\u8ef8\u3068\u77ed\u8ef8\u306e\u9577\u3055\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(AP^2+BP^2\\) \u304a\u3088\u3073 \\(AP \\cdot BP\\) \u3092, \\(OP\\) \u3068 \\(d\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\u70b9 \\(P\\) \u304c\u6955\u5186 \\(E\\) \u5168\u4f53\u3092\u52d5\u304f\u3068\u304d, \\(AP^3+BP^3\\) \u306e\u6700\u5927\u5024\u3068\u6700\u5c0f\u5024\u3092 \\(d\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

    \u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n

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

    \u7126\u70b9 \\(A , B\\) \u304c \\(x\\) \u8ef8\u4e0a\u306b\u3042\u308b\u306e\u3067, \u6955\u5186 \\(E\\) \u306e\u5f0f\u306f\r\n\\[\r\n\\dfrac{x^2}{a^2} +\\dfrac{y^2}{b^2} = 1 \\quad ( a \\gt b \\gt 0 )\r\n\\]\r\n\u3068\u304a\u304f\u3053\u3068\u304c\u3067\u304d\u308b.
    \r\n\u3053\u3053\u3067\u9577\u8ef8\u306e\u9577\u3055\u306f \\(a\\) , \u77ed\u8ef8\u306e\u9577\u3055\u306f \\(b\\) \u3067\u3042\u308a,\r\n\\[\\begin{align}\r\n(a-d)+(a+d) & = 4d \\\\\r\n\\text{\u2234} \\quad a & = 2d\r\n\\end{align}\\]\r\n\u307e\u305f\r\n\\[\\begin{align}\r\n2 \\sqrt{b^2+d^2} & = 4d \\\\\r\n\\text{\u2234} \\quad b^2+d^2 & = 4d^2 \\\\\r\n\\text{\u2234} \\quad b & = \\sqrt{3}d\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, \u9577\u8ef8\u306e\u9577\u3055\u306f \\(\\underline{2d}\\) , \u77ed\u8ef8\u306e\u9577\u3055\u306f \\(\\underline{\\sqrt{3}d}\\) .<\/p>\r\n

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

    \\(P \\ ( X , Y )\\) \u3068\u304a\u304f\u3068\r\n\\[\r\nOP^2 = X^2+Y^2\r\n\\]\r\n\u3053\u308c\u3068, \\(AP+BP = 4d\\) \u3092\u7528\u3044\u308c\u3070\r\n\\[\\begin{align}\r\nAP^2+BP^2 & = (X-d)^2+Y^2+(X+d)^2+Y^2 \\\\\r\n& = 2 \\left( X^2+Y^2+d^2 \\right) = \\underline{2 \\left( OP^2 +d^2 \\right)}\r\n\\end{align}\\]\r\n\u307e\u305f\r\n\\[\\begin{align}\r\n( AP+BP )^2 & = 2 \\left( OP^2+d^2 \\right) +2 AP \\cdot BP = 16d^2 \\\\\r\n\\text{\u2234} \\quad AP \\cdot BP & = \\underline{7d^2 -OP^2}\r\n\\end{align}\\]\r\n

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

    \\[\\begin{align}\r\nAP^3+BP^3 & = (AP+BP)^3 -3 AP \\cdot BP (AP+BP) \\\\\r\n& = 64d^3 -3 \\left( 7d^2 -OP^2 \\right) \\cdot 4d \\\\\r\n& = 12d OP^2 -20d^3\r\n\\end{align}\\]\r\n(1)<\/strong> \u306e\u7d50\u679c\u3088\u308a, \\(3d^2 \\leqq OP^2 \\leqq 4d^2\\) \u306a\u306e\u3067\r\n\\[\r\n16d^3 \\leqq AP^3+BP^3 \\leqq 28d^3\r\n\\]\r\n\u3088\u3063\u3066, \u6700\u5927\u5024\u306f \\(\\underline{28d^3}\\) , \u6700\u5c0f\u5024\u306f \\(\\underline{16d^3}\\) .<\/p>\r\n","protected":false},"excerpt":{"rendered":"\\(d\\) \u3092\u6b63\u306e\u5b9a\u6570\u3068\u3059\u308b. \\(2\\) \u70b9 \\(A \\ ( -d , 0 )\\) , \\(B \\ ( d , 0 )\\) \u304b\u3089\u306e\u8ddd\u96e2\u306e\u548c\u304c \\(4d\\) \u3067\u3042\u308b\u70b9 \\(P\\) \u306e\u8ecc\u8de1\u3068\u3057\u3066\u5b9a\u307e\u308b\u6955\u5186 \\(E\\) \u3092\u8003 … \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":[41],"tags":[144,13],"class_list":["post-82","post","type-post","status-publish","format-standard","hentry","category-tsukuba_r_2011","tag-tsukuba_r","tag-13"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/82","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=82"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/82\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=82"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=82"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=82"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}