{"id":882,"date":"2013-08-19T00:11:08","date_gmt":"2013-08-18T15:11:08","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=882"},"modified":"2021-10-20T14:21:10","modified_gmt":"2021-10-20T05:21:10","slug":"ykr201303","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/ykr201303\/","title":{"rendered":"\u6a2a\u56fd\u5927\u7406\u7cfb2013\uff1a\u7b2c3\u554f"},"content":{"rendered":"
\n

\\(0 \\lt \\theta \\lt \\dfrac{\\pi}{3}\\) \u3092\u6e80\u305f\u3059 \\(\\theta\\) \u306b\u5bfe\u3057, \\(xy\\) \u5e73\u9762\u306e\u7b2c \\(1\\) \u8c61\u9650\u306e\u70b9 P \u304a\u3088\u3073 \\(x\\) \u8ef8\u306e\u6b63\u306e\u90e8\u5206\u306b\u3042\u308b\u70b9 Q \u3092\r\n\\[\r\n\\angle \\text{QOP} = \\theta , \\ \\angle \\text{PQO} = 2 \\theta , \\ \\text{PQ} = 1\r\n\\]\r\n\u3092\u6e80\u305f\u3059\u3088\u3046\u306b\u3068\u308b. PQ \u306e\u4e2d\u70b9\u3092 R \u3068\u3059\u308b. \\(\\theta\\) \u304c \\(0 \\lt \\theta \\lt \\dfrac{\\pi}{3}\\) \u306e\u7bc4\u56f2\u3092\u52d5\u304f\u3068\u304d, P \u306e\u8ecc\u8de1\u3092 \\(C _ 1\\) , R \u306e\u8ecc\u8de1\u3092 \\(C _ 2\\) \u3068\u3059\u308b. \u6b21\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000P , Q , R \u306e\u5ea7\u6a19\u3092 \\(\\theta\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(C _ 1 , C _ 2\\) \u3092\u6c42\u3081, \u305d\u308c\u3089\u3092\u56f3\u793a\u305b\u3088.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\\(C _ 1 , C _ 2\\) \u304a\u3088\u3073 \\(x\\) \u8ef8\u3067\u56f2\u307e\u308c\u308b\u90e8\u5206\u3092 \\(x\\) \u8ef8\u306e\u307e\u308f\u308a\u306b \\(1\\) \u56de\u8ee2\u3057\u3066\u3067\u304d\u308b\u56de\u8ee2\u4f53\u306e\u4f53\u7a4d\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

    \u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n\"yokokoku_r_2013_03_04\"\r\n

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

    Q \\(( q , 0 )\\) \uff08 \\(q \\gt 0\\) \uff09\u3068\u304a\u304f\u3068, \\(\\angle \\text{OQP} = 2 \\theta\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\n\\text{P} & \\ ( q -\\cos 2 \\theta , \\sin 2 \\theta ) \\quad ... [1] , \\\\\r\n\\text{R} & \\ \\left( q -\\dfrac{\\cos 2 \\theta}{2} , \\dfrac{\\sin 2 \\theta}{2} \\right)\r\n\\end{align}\\]\r\n\u3068\u8868\u305b\u308b.\r\n\u307e\u305f, \\(\\angle \\text{POQ} = \\theta\\) \u306a\u306e\u3067, \u5b9f\u6570 \\(p \\ ( p \\gt 0 )\\) \u3092\u7528\u3044\u3066\r\n\\[\r\n\\text{P} \\ ( p \\cos \\theta , p \\sin \\theta ) \\quad ... [2]\r\n\\]\r\n\u3068\u8868\u305b\u308b.
    \r\n[1] [2] \u3092\u6bd4\u8f03\u3057\u3066\r\n\\[\\begin{gather}\r\n\\left\\{\\begin{array}{l} p \\cos \\theta = q -\\cos 2 \\theta \\\\ p \\sin \\theta = \\sin 2 \\theta \\end{array}\\right. \\\\\r\n\\text{\u2234} \\quad p = 2 \\cos \\theta , \\ q = 2 \\cos 2 \\theta +1\r\n\\end{gather}\\]\r\n\u3088\u3063\u3066, \u5404\u70b9\u306e\u5ea7\u6a19\u306f\r\n\\[\\begin{align}\r\n\\text{P} & \\ \\underline{( \\cos 2 \\theta +1 , \\sin 2 \\theta )} \\\\\r\n\\text{Q} & \\ \\underline{( 2 \\cos 2 \\theta +1 , 0 )} \\\\\r\n\\text{R} & \\ \\underline{\\left( \\dfrac{3 \\cos 2 \\theta}{2} +1 , \\dfrac{\\sin 2 \\theta}{2} \\right)}\r\n\\end{align}\\]\r\n

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

    \u6761\u4ef6\u3088\u308a, \\(0 \\lt 2 \\theta \\lt \\dfrac{2 \\pi}{3}\\) \u306a\u306e\u3067,\r\n\\[\r\n-\\dfrac{1}{2} \\lt \\cos 2 \\theta \\lt 1 , \\ 0 \\lt \\sin 2 \\theta \\lt 1\r\n\\]\r\n\u3088\u3063\u3066, \u70b9 P , Q \u306e\u8ecc\u8de1\u3067\u3042\u308b \\(C _ 1 , C _ 2\\) \u306f\r\n\\[\\begin{align}\r\nC _ 1 & \uff1a \\underline{(x-1)^2 +y^2 = 1 \\ \\left( \\dfrac{1}{2} \\lt x \\lt 2 , y \\gt 0 \\right)} , \\\\\r\nC _ 2 & \uff1a \\underline{\\dfrac{4 (x-1)^2}{9} +4 y^2 = 1 \\ \\left( \\dfrac{1}{4} \\lt x \\lt \\dfrac{5}{2} , \\ y \\gt 0 \\right)}\r\n\\end{align}\\]\r\n\u3053\u308c\u3092\u56f3\u793a\u3059\u308b\u3068, \u4e0b\u56f3\u5b9f\u7dda\u90e8\uff08\u25cb\u306f\u9664\u304f\uff09<\/p>\r\n\"yokokoku_r_2013_03_01\"\r\n

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

    \\(C _ 1 , C _ 2\\) \u304a\u3088\u3073 \\(x\\) \u8ef8\u3067\u56f2\u307e\u308c\u308b\u90e8\u5206\u3092 \\(D\\) \u3068\u304a\u304f.
    \r\n(2)<\/strong> \u306e\u7d50\u679c\u3088\u308a, \\(C _ 1 , C _ 2\\) \u306e\u5f0f\u304b\u3089 \\(y\\) \u3092\u6d88\u53bb\u3059\u308b\u3068\r\n\\[\\begin{align}\r\n\\dfrac{4 (x-1)^2}{9} +4 \\left\\{ 1 -(x-1)^2 \\right\\} & = 1 \\\\\r\n32 (x-1)^2 & = 27 \\\\\r\n\\text{\u2234} \\quad x = 1 \\pm \\dfrac{3 \\sqrt{6}}{8} &\r\n\\end{align}\\]\r\n\\(\\sqrt{6} \\gt 2\\) \u3088\u308a, \\(\\dfrac{3 \\sqrt{6}}{8} \\gt \\dfrac{3}{4}\\) \u306a\u306e\u3067, \\(C _ 1 , C _ 2\\) \u306e\u4ea4\u70b9\u306e \\(x\\) \u5ea7\u6a19\u306f \\(1 +\\dfrac{3 \\sqrt{6}}{8}\\) \u3067\u3042\u308b.
    \r\n\u3057\u305f\u304c\u3063\u3066, \\(D\\) \u306f\u4e0b\u56f3\u659c\u7dda\u90e8\u306e\u3088\u3046\u306b\u306a\u308b.<\/p>\r\n\"yokokoku_r_2013_03_03\"\r\n

    \u3088\u3063\u3066, \u6c42\u3081\u308b\u4f53\u7a4d \\(V\\) \u306f\r\n\\[\\begin{align}\r\nV & = \\pi \\displaystyle\\int _ {1 +\\frac{3 \\sqrt{6}}{8}}^{\\frac{5}{2}} \\left\\{ \\dfrac{1}{4} -\\dfrac{(x-1)^2}{9} \\right\\} \\, dx \\\\\r\n& \\qquad -\\pi \\displaystyle\\int _ {1 +\\frac{3 \\sqrt{6}}{8}}^{2} \\left\\{ 1 -(x-1)^2 \\right\\} \\, dx \\\\\r\n& = \\pi \\displaystyle\\int _ {\\frac{3 \\sqrt{6}}{8}}^{\\frac{3}{2}} \\left( \\dfrac{1}{4} -\\dfrac{x^2}{9} \\right) \\, dx \\\\\r\n& \\qquad -\\pi \\displaystyle\\int _ {\\frac{3 \\sqrt{6}}{8}}^{1} \\left( 1 -x^2 \\right) \\, dx \\\\\r\n& = \\pi \\left[ \\dfrac{x}{4} -\\dfrac{x^3}{27} \\right] _ {\\frac{3 \\sqrt{6}}{8}}^{\\frac{3}{2}} -\\pi \\left[ x -\\dfrac{x^3}{3} \\right] _ {\\frac{3 \\sqrt{6}}{8}}^{1} \\\\\r\n& = \\pi \\left( \\dfrac{3}{8} -\\dfrac{1}{8} \\right) -\\sqrt{6} \\pi \\left( \\dfrac{3}{32} -\\dfrac{3}{256} \\right) \\\\\r\n& \\qquad -\\dfrac{2 \\pi}{3} +\\sqrt{6} \\pi \\left( \\dfrac{3}{8} -\\dfrac{27}{256} \\right) \\\\\r\n& = \\dfrac{3 \\sqrt{6} \\pi}{16}-\\dfrac{5 \\pi}{12} \\\\\r\n& = \\underline{\\dfrac{\\left( 9 \\sqrt{6} -20 \\right) \\pi}{48}}\r\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\\(0 \\lt \\theta \\lt \\dfrac{\\pi}{3}\\) \u3092\u6e80\u305f\u3059 \\(\\theta\\) \u306b\u5bfe\u3057, \\(xy\\) \u5e73\u9762\u306e\u7b2c \\(1\\) \u8c61\u9650\u306e\u70b9 P \u304a\u3088\u3073 \\(x\\) \u8ef8\u306e\u6b63\u306e\u90e8\u5206\u306b\u3042\u308b\u70b9 Q \u3092 \\[ … \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":[104],"tags":[111,9],"class_list":["post-882","post","type-post","status-publish","format-standard","hentry","category-yokokoku_r_2013","tag-111","tag-yokokoku_r"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/882","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=882"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/882\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=882"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=882"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=882"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}