{"id":924,"date":"2013-09-04T21:47:14","date_gmt":"2013-09-04T12:47:14","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=924"},"modified":"2021-09-30T00:03:19","modified_gmt":"2021-09-29T15:03:19","slug":"kbr201306","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/kbr201306\/","title":{"rendered":"\u7b51\u6ce2\u5927\u7406\u7cfb2013\uff1a\u7b2c6\u554f"},"content":{"rendered":"
\n

\u6955\u5186 \\(C : \\ \\dfrac{x^2}{16} +\\dfrac{y^2}{9} = 1\\) \u306e, \u76f4\u7dda \\(y = mx\\) \u3068\u5e73\u884c\u306a \\(2\\) \u63a5\u7dda\u3092 \\(\\ell _ 1 , \\ell' _ 1\\) \u3068\u3057, \\(\\ell _ 1 , \\ell' _ 1\\) \u306b\u76f4\u4ea4\u3059\u308b \\(C\\) \u306e \\(2\\) \u63a5\u7dda\u3092 \\(\\ell _ 2 , \\ell' _ 2\\) \u3068\u3059\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(\\ell _ 1 , \\ell' _ 1\\) \u306e\u65b9\u7a0b\u5f0f\u3092 \\(m\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(\\ell _ 1\\) \u3068 \\(\\ell' _ 1\\) \u306e\u8ddd\u96e2 \\(d _ 1\\) \u304a\u3088\u3073 \\(\\ell _ 2\\) \u3068 \\(\\ell' _ 2\\) \u306e\u8ddd\u96e2 \\(d _ 2\\) \u3092\u305d\u308c\u305e\u308c \\(m\\) \u3092\u7528\u3044\u3066\u8868\u305b. \u305f\u3060\u3057, \u5e73\u884c\u306a \\(2\\) \u76f4\u7dda \\(\\ell , \\ell'\\) \u306e\u8ddd\u96e2 \\(d\\) \u3068\u306f, \\(\\ell\\) \u4e0a\u306e \\(1\\) \u70b9\u3068\u76f4\u7dda \\(\\ell'\\) \u306e\u8ddd\u96e2\u3067\u3042\u308b.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\\((d _ 1)^2+(d _ 2)^2\\) \u306f \\(m\\) \u306b\u3088\u3089\u305a\u4e00\u5b9a\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n

  4. (4)<\/strong>\u3000\\(\\ell _ 1 , \\ell' _ 1 , \\ell _ 2 , \\ell' _ 2\\) \u3067\u56f2\u307e\u308c\u308b\u9577\u65b9\u5f62\u306e\u9762\u7a4d \\(S\\) \u3092 \\(d _ 1\\) \u3092\u7528\u3044\u3066\u8868\u305b. \u3055\u3089\u306b \\(m\\) \u304c\u5909\u5316\u3059\u308b\u3068\u304d, \\(S\\) \u306e\u6700\u5927\u5024\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

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

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

    \u76f4\u7dda \\(\\ell _ 1 : \\ y = mx+n\\) \u3068\u304a\u304f.
    \r\n\u6955\u5186 \\(C\\) \u306e\u5f0f\u3068\u76f4\u7dda \\(\\ell _ 1\\) \u306e\u5f0f\u304b\u3089, \\(y\\) \u3092\u6d88\u53bb\u3059\u308b\u3068\r\n\\[\\begin{gather}\r\n9x^2 +16(mx+n)^2 = 144 \\\\\r\n\\text{\u2234} \\quad ( 16m^2+9 )x^2 +32mnx +16( n^2-9 ) = 0\r\n\\end{gather}\\]\r\n\u3053\u306e \\(x\\) \u306e \\(2\\) \u6b21\u65b9\u7a0b\u5f0f\u304c, \u91cd\u89e3\u3092\u3082\u3066\u3070 \\(C\\) \u3068 \\(\\ell _ 1\\) \u304c\u63a5\u3059\u308b\u306e\u3067, \u5224\u5225\u5f0f \\(D\\) \u306b\u3064\u3044\u3066\r\n\\[\\begin{gather}\r\n\\dfrac{D}{4} = (16mn)^2 -16 ( 16m^2+9 ) ( n^2-9 ) = 0 \\\\\r\nn^2 -16 m^2 -9 = 0 \\\\\r\n\\text{\u2234} \\quad n = \\pm \\sqrt{16m^2+9}\r\n\\end{gather}\\]\r\n\u3088\u3063\u3066, \\(\\ell _ 1 , {\\ell _ 1}'\\) \u306e\u5f0f\u306f\r\n\\[\r\n\\underline{y = mx \\pm \\sqrt{16m^2+9}}\r\n\\]\r\n

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

    \\(\\ell _ 1\\) \u4e0a\u306e\u70b9 \\(\\left( t , mt +\\sqrt{16m^2+9} \\right)\\) \u3068 \\({\\ell _ 1}'\\) \u3068\u306e\u8ddd\u96e2\u3092\u6c42\u3081\u308c\u3070\u3088\u3044\u306e\u3067\r\n\\[\\begin{align}\r\nd _ 1 & = \\dfrac{\\left| -2 \\sqrt{16m^2+9} \\right|}{\\sqrt{m^2+1}} \\\\\r\n& = \\underline{\\dfrac{2 \\sqrt{16m^2+9}}{\\sqrt{m^2+1}}} \\quad ... [1]\r\n\\end{align}\\]\r\n\\(\\ell _ 1 \\perp \\ell _ 2\\) \u306a\u306e\u3067, \\(\\ell _ 2\\) \u306e\u50be\u304d\u306f \\(-\\dfrac{1}{m}\\) \u3067\u3042\u308b.
    \r\n\\(\\ell _ 2 , {\\ell _ 2}'\\) \u306b\u3064\u3044\u3066\u3082 \\(\\ell _ 1 , {\\ell _ 1}'\\) \u3068\u540c\u69d8\u306b\u8003\u3048\u308b\u3053\u3068\u304c\u3067\u304d, [1] \u306b\u304a\u3044\u3066 \\(m \\rightarrow -\\dfrac{1}{m}\\) \u3068\u7f6e\u63db\u3048\u308c\u3070\r\n\\[\\begin{align}\r\nd _ 2 & = \\dfrac{2 \\sqrt{16 \\left( -\\frac{1}{m} \\right)^2+9}}{\\sqrt{\\left( -\\frac{1}{m} \\right)^2 +1}} \\\\\r\n& = \\underline{\\dfrac{2 \\sqrt{9m^2+16}}{\\sqrt{m^2+1}}}\r\n\\end{align}\\]\r\n

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

    (2)<\/strong> \u306e\u7d50\u679c\u3092\u7528\u3044\u308c\u3070\r\n\\[\\begin{align}\r\n(d _ 1)^2 +(d _ 2)^2 & = \\dfrac{4 ( 16m^2+9 )}{m^2+1} +\\dfrac{4 ( 9m^2+16 )}{m^2+1} \\\\\r\n& = 4 \\cdot 25 = 100\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, \\(m\\) \u306e\u5024\u306b\u3088\u3089\u305a\u4e00\u5b9a\u3067\u3042\u308b.<\/p>\r\n

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

    (3)<\/strong> \u306e\u7d50\u679c\u3088\u308a\r\n\\[\r\nd _ 2 = \\sqrt{100 -(d _ 1)^2}\r\n\\]\r\n\u3067\u3042\u308a, \\(0 \\leqq d _ 1 \\leqq 10\\) ... [2] \u3067\u3042\u308b.
    \r\n\u3086\u3048\u306b\r\n\\[\\begin{align}\r\nS & = d _ 1 d _ 2 \\\\\r\n& = \\underline{d _ 1 \\sqrt{100 -(d _ 1)^2}}\r\n\\end{align}\\]\r\n\u3053\u3053\u3067, \u76f8\u52a0\u76f8\u4e57\u5e73\u5747\u306e\u95a2\u4fc2\u3092\u7528\u3044\u308b\u3068\r\n\\[\\begin{align}\r\nS & \\leqq \\dfrac{(d _ 1)^2 +\\left\\{ 100 -(d _ 1)^2 \\right\\}}{2} \\\\\r\n& = 50\r\n\\end{align}\\]\r\n\u7b49\u53f7\u6210\u7acb\u306f, \\((a _ 1)^2 = 100 -(d _ 1)^2\\) \u3059\u306a\u308f\u3061, \\(d _ 1 = 5\\sqrt{2}\\) \u306e\u3068\u304d\u3067, [2] \u3092\u307f\u305f\u3057\u3066\u3044\u308b.
    \r\n\u3088\u3063\u3066, \\(S\\) \u306e\u6700\u5927\u5024\u306f\r\n\\[\r\n\\underline{50}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u6955\u5186 \\(C : \\ \\dfrac{x^2}{16} +\\dfrac{y^2}{9} = 1\\) \u306e, \u76f4\u7dda \\(y = mx\\) \u3068\u5e73\u884c\u306a \\(2\\) \u63a5\u7dda\u3092 \\(\\ell _ 1 , \\ell' _ 1\\) \u3068\u3057, … \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":[102],"tags":[144,111],"class_list":["post-924","post","type-post","status-publish","format-standard","hentry","category-tsukuba_r_2013","tag-tsukuba_r","tag-111"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/924","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=924"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/924\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=924"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=924"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=924"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}