{"id":1852,"date":"2021-09-05T20:43:45","date_gmt":"2021-09-05T11:43:45","guid":{"rendered":"https:\/\/www.roundown.net\/nyushi\/?p=1852"},"modified":"2021-09-08T13:17:57","modified_gmt":"2021-09-08T04:17:57","slug":"ngr201602","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/ngr201602\/","title":{"rendered":"\u540d\u53e4\u5c4b\u5927\u7406\u7cfb2016\uff1a\u7b2c2\u554f"},"content":{"rendered":"
\n

\\(2\\) \u3064\u306e\u5186 \\(C : \\ (x-1)^2 +y^2 = 1\\) \u3068 \\(D : \\ (x+2)^2 +y^2 = 7^2\\) \u3092\u8003\u3048\u308b. \u307e\u305f\u539f\u70b9\u3092 O \\(( 0 , 0 )\\) \u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \u6b21\u306e\u554f\u306b\u7b54\u3048\u3088.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\u5186 \\(C\\) \u4e0a\u306b, \\(y\\) \u5ea7\u6a19\u304c\u6b63\u3067\u3042\u308b\u3088\u3046\u306a\u70b9 P \u3092\u3068\u308a, \\(x\\) \u8ef8\u306e\u6b63\u306e\u90e8\u5206\u3068\u7dda\u5206 OP \u306e\u306a\u3059\u89d2\u3092 \\(\\theta\\) \u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \u70b9 P \u306e\u5ea7\u6a19\u3068\u7dda\u5206 OP \u306e\u9577\u3055\u3092 \\(\\theta\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000(1)<\/strong> \u3067\u3068\u3063\u305f\u70b9 P \u3092\u56fa\u5b9a\u3057\u305f\u307e\u307e, \u70b9 Q \u304c\u5186 \\(D\\) \u4e0a\u3092\u52d5\u304f\u3068\u304d, \\(\\triangle \\text{OPQ}\\) \u306e\u9762\u7a4d\u304c\u6700\u5927\u306b\u306a\u308b\u3068\u304d\u306e Q \u306e\u5ea7\u6a19\u3092 \\(\\theta\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\u70b9 P \u304c\u5186 \\(C\\) \u4e0a\u3092\u52d5\u304d, \u70b9 Q \u304c\u5186 \\(D\\) \u4e0a\u3092\u52d5\u304f\u3068\u304d, \\(\\triangle \\text{OPQ}\\) \u306e\u9762\u7a4d\u306e\u6700\u5927\u5024\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n

    \u305f\u3060\u3057, (2)<\/strong> , (3)<\/strong> \u306b\u304a\u3044\u3066\u306f, \\(3\\) \u70b9 O , P , Q \u304c\u540c\u4e00\u76f4\u7dda\u4e0a\u306b\u3042\u308b\u3068\u304d\u306f, \\(\\triangle \\text{OPQ}\\) \u306e\u9762\u7a4d\u306f \\(0\\) \u3067\u3042\u308b\u3068\u3059\u308b.<\/p>\r\n

    \r\n\r\n

    \u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n

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

    \\(r = \\text{OP}\\) \u3068\u304a\u3051\u3070, P \u306e\u5ea7\u6a19\u306f \\(( r \\cos \\theta , r \\sin \\theta )\\) \u3068\u3042\u3089\u308f\u305b\u308b.
    \r\n\u307e\u305f, P \u306f\u7b2c \\(1\\) \u8c61\u9650\u306b\u3042\u308b\u306e\u3067, \\(0 \\lt \\theta \\lt \\dfrac{\\pi}{2}\\) ... [1] .
    \r\n\u5186 \\(C\\) \u306e\u4e2d\u5fc3\u3092 A \\(( 1 , 0 )\\) \u3068\u304a\u3051\u3070, \\(\\triangle \\text{AOP}\\) \u306f\u4e8c\u7b49\u8fba\u4e09\u89d2\u5f62\u306a\u306e\u3067\r\n\\[\r\n\\text{OP} = r = \\underline{2 \\cos \\theta} \\ .\r\n\\]\r\n\u307e\u305f\r\n\\[\r\n\\text{P} \\ \\underline{\\left( \\cos 2 \\theta +1 , \\sin 2 \\theta \\right)} \\ .\r\n\\]\r\n

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

    \u5186 \\(D\\) \u306e\u4e2d\u5fc3\u3092 B \\(( -2 , 0 )\\) \u3068\u304a\u304f.
    \r\n\\(\\triangle \\text{OPQ}\\) \u306e\u9762\u7a4d\u304c\u6700\u5927\u3068\u306a\u308b\u306e\u306f, Q \u304c\u76f4\u7dda OP \u304b\u3089\u6700\u3082\u96e2\u308c\u305f\u3068\u304d, \u3064\u307e\u308a Q \u304c OP \u3088\u308a B \u5074\u306b\u3042\u308a, OP \u3068\u50be\u304d\u306e\u7b49\u3057\u3044 \\(D\\) \u306e\u63a5\u7dda\u3068\u306e\u63a5\u70b9\u3068\u306a\u3063\u305f\u3068\u304d\u3067\u3042\u308b.
    \r\n\u3053\u306e\u3068\u304d, OQ \u304c \\(x\\) \u8ef8\u6b63\u65b9\u5411\u3068\u306a\u3059\u89d2\u306f \\(\\theta +\\dfrac{\\pi}{2}\\) \u3067\u3042\u308b.
    \r\n\u3088\u3063\u3066, \u6c42\u3081\u308b Q \u306e\u5ea7\u6a19\u306f\r\n\\[\r\n\\left( -2 +7 \\cos \\left( \\theta +\\dfrac{\\pi}{2} \\right) , 7 \\sin \\left( \\theta +\\dfrac{\\pi}{2} \\right) \\right)\r\n\\]\r\n\u3059\u306a\u308f\u3061\r\n\\[\r\n\\underline{\\left( -2 -7 \\sin \\theta , 7 \\cos \\theta \\right)} \\ .\r\n\\]\r\n

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

    (2)<\/strong> \u306e\u3088\u3046\u306b, \u70b9 P \u3092\u56fa\u5b9a\u3057\u3066\u8003\u3048\u308b.
    \r\n\u3053\u306e\u3068\u304d, \u76f4\u7dda OP \u3068 BQ \u306f\u76f4\u4ea4\u3057, \u305d\u306e\u4ea4\u70b9\u3092 E \u3068\u304a\u304f\u3068\r\n\\[\r\n\\text{BE} = 2 \\sin \\theta \\ .\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(\\triangle \\text{OPQ}\\) \u306e\u9762\u7a4d \\(S\\) \u306f\r\n\\[\\begin{align}\r\nS & = \\dfrac{1}{2} \\left( 2 \\cos \\theta \\right) \\left( 7 +2 \\sin \\theta \\right) \\\\\r\n& = \\cos \\theta \\left( 7 +2 \\sin \\theta \\right) \\ .\r\n\\end{align}\\]\r\n\u3064\u3065\u3044\u3066, \u70b9 P \u3092\u52d5\u304b\u3057\u3066\u8003\u3048\u308b\u3068, \\(f( \\theta ) = S\\) \u3068\u304a\u3044\u3066, [1] \u306e\u7bc4\u56f2\u306b\u304a\u3051\u308b, \\(f( \\theta )\\) \u306e\u6700\u5927\u5024\u3092\u6c42\u3081\u308c\u3070\u3088\u3044.\r\n\\[\\begin{align}\r\nf'( \\theta ) & = -\\sin \\theta \\left( 7 +2 \\sin \\theta \\right) +2 \\cos^2 \\theta \\\\\r\n& = -4 \\sin^2 \\theta -7 \\sin \\theta +2 \\\\\r\n& = -\\left( 4 \\sin \\theta -1 \\right) \\left( \\sin \\theta +2 \\right) \\ .\r\n\\end{align}\\]\r\n[1] \u306b\u304a\u3044\u3066, \\(\\sin \\theta\\) \u306f\u5358\u8abf\u5897\u52a0\u306a\u306e\u3067, \\(f( \\theta )\\) \u306e\u5897\u6e1b\u306f\u4e0b\u8868\u306e\u3068\u304a\u308a.\r\n\\[\r\n\\begin{array}{c|ccccc} \\theta & ( 0 ) & \\cdots & \\alpha & \\cdots & \\left( \\dfrac{\\pi}{2} \\right) \\\\ \\hline f'( \\theta ) & & + & 0 & - & \\\\ \\hline f( \\theta ) & ( 7 ) & \\nearrow & \\text{\u6700\u5927} & \\searrow & ( 0 ) \\end{array}\r\n\\]\r\n\u305f\u3060\u3057, \\(\\sin \\alpha = \\dfrac{1}{4} \\ \\left( 0 \\lt \\alpha \\lt \\dfrac{\\pi}{2} \\right)\\) \u3068\u304a\u3044\u305f.
    \r\n\\(\\cos \\alpha = \\sqrt{1 -\\left( \\dfrac{1}{4} \\right)^2} = \\dfrac{\\sqrt{15}}{4}\\) \u306a\u306e\u3067, \u6c42\u3081\u308b\u6700\u5927\u5024\u306f\r\n\\[\r\nf( \\alpha ) = \\dfrac{\\sqrt{15}}{4} \\left( 7 +2 \\cdot \\dfrac{1}{4} \\right) = \\underline{\\dfrac{15 \\sqrt{15}}{8}} \\ .\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(2\\) \u3064\u306e\u5186 \\(C : \\ (x-1)^2 +y^2 = 1\\) \u3068 \\(D : \\ (x+2)^2 +y^2 = 7^2\\) \u3092\u8003\u3048\u308b. \u307e\u305f\u539f\u70b9\u3092 O \\(( 0 , 0 )\\) \u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \u6b21\u306e\u554f … \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":[156],"tags":[143,162],"class_list":["post-1852","post","type-post","status-publish","format-standard","hentry","category-nagoya_r_2016","tag-nagoya_r","tag-162"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1852","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=1852"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1852\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1852"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1852"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1852"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}