{"id":540,"date":"2012-12-27T21:33:55","date_gmt":"2012-12-27T12:33:55","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=540"},"modified":"2021-10-31T07:26:25","modified_gmt":"2021-10-30T22:26:25","slug":"tkb201004","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tkb201004\/","title":{"rendered":"\u6771\u5927\u6587\u7cfb2010\uff1a\u7b2c4\u554f"},"content":{"rendered":"
\n

\\(C\\) \u3092\u534a\u5f84 \\(1\\) \u306e\u5186\u5468\u3068\u3057, A \u3092 \\(C\\) \u4e0a\u306e \\(1\\) \u70b9\u3068\u3059\u308b.\r\n\\(3\\) \u70b9 P , Q , R \u304c A \u3092\u6642\u523b \\(t = 0\\) \u306b\u51fa\u767a\u3057, \\(C\\) \u4e0a\u3092\u5404\u3005\u4e00\u5b9a\u306e\u901f\u3055\u3067, P , Q \u306f\u53cd\u6642\u8a08\u56de\u308a\u306b, R \u306f\u6642\u8a08\u56de\u308a\u306b, \u6642\u523b \\(t = 2\\pi\\) \u307e\u3067\u52d5\u304f.\r\nP , Q , R \u306e\u901f\u3055\u306f, \u305d\u308c\u305e\u308c \\(m , 1 , 2\\) \u3067\u3042\u308b\u3068\u3059\u308b.\uff08\u3057\u305f\u304c\u3063\u3066, Q\u306f \\(C\\) \u3092\u3061\u3087\u3046\u3069\u4e00\u5468\u3059\u308b. \uff09\u305f\u3060\u3057, \\(m\\) \u306f \\(1 \\leqq m \\leqq 10\\) \u3092\u307f\u305f\u3059\u6574\u6570\u3067\u3042\u308b.\r\n\\(\\triangle \\text{PQR}\\) \u304c PR \u3092\u659c\u8fba\u3068\u3059\u308b\u76f4\u89d2\u4e8c\u7b49\u8fba\u4e09\u89d2\u5f62\u3068\u306a\u308b\u3088\u3046\u306a\u901f\u3055 \\(m\\) \u3068\u6642\u523b \\(t\\) \u306e\u7d44\u3092\u3059\u3079\u3066\u6c42\u3081\u3088.<\/p>\r\n

\r\n\r\n

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

\\(xy\\) \u5e73\u9762\u4e0a\u3067, A \\((1,0)\\) \u3068\u304a\u304f\u3068\r\n\\[\r\n\\text{P} ( \\cos mt , \\sin mt ) , \\ \\text{Q} ( \\cos t , \\sin t ) , \\ \\text{R} ( \\cos (-2t) , \\sin (-2t) )\r\n\\]\r\n\u3068\u8868\u305b\u308b.\r\n\\(\\triangle \\text{PQR}\\) \u304c PR \u3092\u659c\u8fba\u3068\u3059\u308b\u76f4\u89d2\u4e8c\u7b49\u8fba\u4e09\u89d2\u5f62\u3068\u306a\u308b\u306e\u306f\r\n\\[\r\n\\left\\{\\begin{array}{ll} \\angle \\text{POR} = \\pi & \\quad ... [1] \\\\ \\angle \\text{ROQ} = \\dfrac{\\pi}{2} & \\quad ... [2] \\end{array}\\right.\r\n\\]\r\n\u304c\u6210\u308a\u7acb\u3064\u3068\u304d\u3067\u3042\u308b.<\/p>\r\n\"todai2010_05_01\"\r\n[2] \u3088\u308a\r\n\\[\r\n\\left| t -(-2t) -k \\pi \\right| = \\left| 3t - k \\pi \\right| = \\dfrac{\\pi}{2} \\quad ( k \\text{\u306f\u6574\u6570} )\r\n\\]\r\n\\(0 \\leqq t \\leqq 2 \\pi\\) \u3088\u308a \\(0 \\leqq 3t \\leqq 6 \\pi\\) \u306a\u306e\u3067\r\n\\[\r\n3t = \\dfrac{\\pi}{2} +k \\pi \\ ( k = 0 , 1 , \\cdots , 5 ) \\quad ... [3]\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, [1] [3] \u3092\u307f\u305f\u3059 \\((m,t)\\) \u306e\u7d44\u3092\u6c42\u3081\u308c\u3070\u3088\u3044.
\r\n\u4ee5\u4e0b\u3067\u306f, \\(k = 0 , 1 , \\cdots , 5\\) \u305d\u308c\u305e\u308c\u306e\u5834\u5408\u306b\u3064\u3044\u3066, \u6761\u4ef6\u3092\u307f\u305f\u3059\u6574\u6570 \\(m\\) \u3092\u6c42\u3081\u308b.<\/p>\r\n

    \r\n

  1. 1*<\/strong>\u3000\\(k=0\\) \u306e\u3068\u304d
    \r\n[3] \u3088\u308a,\u3000\\(t = \\dfrac{\\pi}{6}\\) .
    \r\n\u3053\u306e\u3068\u304d, \\(\\angle \\text{AOR} = -\\dfrac{\\pi}{3}\\) \u306a\u306e\u3067 [1] \u3088\u308a,\u3000\\(\\angle \\text{AOP} = \\dfrac{2}{3}\\pi\\) .
    \r\n\u3086\u3048\u306b\r\n\\[\r\nm \\cdot \\dfrac{\\pi}{6} = \\left( \\dfrac{2}{3} +2l \\right) \\pi \\Longleftrightarrow m = 4 +12l \\quad ( l \\text{\u306f\u6574\u6570} )\r\n\\]\r\n\\(1 \\leqq m \\leqq 10\\) \u306a\u306e\u3067\r\n\\[\r\nm = 4\r\n\\]<\/li>\r\n

  2. 2*<\/strong>\u3000\\(k=1\\) \u306e\u3068\u304d
    \r\n[3] \u3088\u308a,\u3000\\(t = \\dfrac{3}{2} \\pi \\cdot \\dfrac{1}{3} = \\dfrac{\\pi}{2}\\) .
    \r\n\u3053\u306e\u3068\u304d, \\(\\angle \\text{AOR} = -\\pi\\) \u306a\u306e\u3067 [1] \u3088\u308a, \\(\\angle \\text{AOP} = 0\\) .
    \r\n\u3086\u3048\u306b\r\n\\[\r\nm \\cdot \\dfrac{\\pi}{2} = 2l \\pi \\Longleftrightarrow m = 4l \\quad ( l \\text{\u306f\u6574\u6570} )\r\n\\]\r\n\\(1 \\leqq m \\leqq 10\\) \u306a\u306e\u3067\r\n\\[\r\nm = 4 , 8\r\n\\]<\/li>\r\n

  3. 3*<\/strong>\u3000\\(k=2\\) \u306e\u3068\u304d
    \r\n[3] \u3088\u308a,\u3000\\(t = \\dfrac{5}{2} \\pi \\cdot \\dfrac{1}{3} = \\dfrac{5}{6} \\pi\\) .
    \r\n\u3053\u306e\u3068\u304d, \\(\\angle \\text{AOR} = \\dfrac{\\pi}{3}\\) \u306a\u306e\u3067 [1] \u3088\u308a, \\(\\angle \\text{AOP} = \\dfrac{4}{3}\\pi\\) .
    \r\n\u3086\u3048\u306b\r\n\\[\r\nm \\cdot \\dfrac{5}{6} \\pi = \\left( \\dfrac{4}{3} +2l \\right) \\pi \\Longleftrightarrow m = \\dfrac{4}{5}(2+3l) \\quad ( l \\text{\u306f\u6574\u6570} )\r\n\\]\r\n\\(1 \\leqq m \\leqq 10\\) \u306a\u306e\u3067\r\n\\[\r\nm = 4\r\n\\]<\/li>\r\n

  4. 4*<\/strong>\u3000\\(k=3\\) \u306e\u3068\u304d
    \r\n[3] \u3088\u308a,\u3000\\(t = \\dfrac{7}{2} \\pi \\cdot \\dfrac{1}{3} = \\dfrac{7}{6} \\pi\\) .
    \r\n\u3053\u306e\u3068\u304d, \\(\\angle \\text{AOR} = -\\dfrac{\\pi}{3}\\) \u306a\u306e\u3067 [1] \u3088\u308a, \\(\\angle \\text{AOP} = \\dfrac{2}{3}\\pi\\) .
    \r\n\u3086\u3048\u306b\r\n\\[\r\nm \\cdot \\dfrac{7}{6} \\pi = \\left( \\dfrac{2}{3} +2l \\right) \\pi \\Longleftrightarrow m = \\dfrac{4}{7}(1+3l) \\quad ( l \\text{\u306f\u6574\u6570} )\r\n\\]\r\n\\(1 \\leqq m \\leqq 10\\) \u306a\u306e\u3067\r\n\\[\r\nm = 4\r\n\\]<\/li>\r\n

  5. 5*<\/strong>\u3000\\(k=4\\) \u306e\u3068\u304d
    \r\n[3] \u3088\u308a,\u3000\\(t = \\dfrac{9}{2} \\pi \\cdot \\dfrac{1}{3} = \\dfrac{3}{2} \\pi\\) .
    \r\n\u3053\u306e\u3068\u304d, \\(\\angle \\text{AOR} = -\\pi\\) \u306a\u306e\u3067[1]\u3088\u308a, \\(\\angle \\text{AOP} = 0\\) .
    \r\n\u3086\u3048\u306b\r\n\\[\r\nm \\cdot \\dfrac{3}{2} \\pi = 2l \\pi \\Longleftrightarrow m = \\dfrac{4l}{3} \\quad ( l \\text{\u306f\u6574\u6570} )\r\n\\]\r\n\\(1 \\leqq m \\leqq 10\\) \u306a\u306e\u3067\r\n\\[\r\nm = 4 , 8\r\n\\]<\/li>\r\n

  6. 6*<\/strong>\u3000\\(k=5\\) \u306e\u3068\u304d
    \r\n[3] \u3088\u308a,\u3000\\(t = \\dfrac{11}{2} \\pi \\cdot \\dfrac{1}{3} = \\dfrac{11}{6} \\pi\\) .
    \r\n\u3053\u306e\u3068\u304d, \\(\\angle \\text{AOR} = \\dfrac{\\pi}{3}\\) \u306a\u306e\u3067 [1] \u3088\u308a, \\(\\angle \\text{AOP} = \\dfrac{4}{3}\\pi\\) .
    \r\n\u3086\u3048\u306b\r\n\\[\r\nm \\cdot \\dfrac{11}{6} \\pi = \\left( \\dfrac{4}{3} +2l \\right) \\pi \\Longleftrightarrow m = \\dfrac{4}{11}(2+3l) \\quad ( l \\text{\u306f\u6574\u6570} )\r\n\\]\r\n\\(1 \\leqq m \\leqq 10\\) \u306a\u306e\u3067\r\n\\[\r\nm = 4\r\n\\]<\/li>\r\n<\/ol>\r\n

    \u4ee5\u4e0a\u3088\u308a, \u6c42\u3081\u308b\u7d44\u306f\r\n\\[\r\n( m , t ) = \\underline{\\left( 4, \\dfrac{\\pi}{6} \\right) , \\left( 4, \\dfrac{\\pi}{2} \\right) , \\left( 8, \\dfrac{\\pi}{2} \\right) , \\left( 4, \\dfrac{5}{6}\\pi \\right) , \\left( 4, \\dfrac{7}{6}\\pi \\right) , \\left( 4, \\dfrac{3}{2}\\pi \\right) , \\left( 8, \\dfrac{3}{2}\\pi \\right) , \\left( 4, \\dfrac{11}{6}\\pi \\right)}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(C\\) \u3092\u534a\u5f84 \\(1\\) \u306e\u5186\u5468\u3068\u3057, A \u3092 \\(C\\) \u4e0a\u306e \\(1\\) \u70b9\u3068\u3059\u308b. \\(3\\) \u70b9 P , Q , R \u304c A \u3092\u6642\u523b \\(t = 0\\) \u306b\u51fa\u767a\u3057, \\(C\\) \u4e0a\u3092\u5404\u3005\u4e00\u5b9a\u306e\u901f\u3055\u3067, P … \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":[80],"tags":[149,14],"class_list":["post-540","post","type-post","status-publish","format-standard","hentry","category-tokyo_b_2010","tag-tokyo_b","tag-14"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/540","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=540"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/540\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=540"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=540"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=540"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}