{"id":1199,"date":"2015-07-29T10:07:32","date_gmt":"2015-07-29T01:07:32","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=1199"},"modified":"2021-10-23T03:00:09","modified_gmt":"2021-10-22T18:00:09","slug":"wsr201405","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/wsr201405\/","title":{"rendered":"\u65e9\u7a32\u7530\u7406\u5de52014\uff1a\u7b2c5\u554f"},"content":{"rendered":"
\n

O \u3092\u539f\u70b9\u3068\u3059\u308b\u5ea7\u6a19\u5e73\u9762\u4e0a\u306b\r\n\\[\r\n\\text{\u653e\u7269\u7dda} \\ C _ 1 : \\ y = x^2 , \\ \\text{\u5186} \\ C _ 2 : \\ x^2 +(y-a)^2 = 1 \\quad ( a \\geqq 0 )\r\n\\]\r\n\u304c\u3042\u308b. \\(C _ 2\\) \u306e\u70b9 \\(( 0 , a+1 )\\) \u306b\u304a\u3051\u308b\u63a5\u7dda\u3068 \\(C _ 1\\) \u304c \\(2\\) \u70b9 A , B \u3067\u4ea4\u308f\u308a, \u25b3OAB \u304c \\(C _ 2\\) \u306b\u5916\u63a5\u3057\u3066\u3044\u308b\u3068\u3059\u308b. \u6b21\u306e\u554f\u306b\u7b54\u3048\u3088.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(a\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\u70b9 \\(( s , t )\\) \u3092 \\(( -1 , a ) , ( 1 , a ) , ( 0 , a-1 )\\) \u3068\u7570\u306a\u308b \\(C _ 2\\) \u4e0a\u306e\u70b9\u3068\u3059\u308b. \u305d\u3057\u3066\u70b9 \\(( s , t )\\) \u306b\u304a\u3051\u308b \\(C _ 2\\) \u306e\u63a5\u7dda\u3068 \\(C _ 1\\) \u3068\u306e \\(2\\) \u3064\u306e\u4ea4\u70b9\u3092 P \\(( \\alpha , {\\alpha}^2 )\\) , Q \\(( \\beta , {\\beta}^2 )\\) \u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \\(( \\alpha -\\beta )^2 -{\\alpha}^2 {\\beta}^2\\) \u306f \\(s, t\\) \u306b\u3088\u3089\u306a\u3044\u5b9a\u6570\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000(2)<\/strong> \u306b\u304a\u3044\u3066, \u70b9 P \\(( \\alpha , {\\alpha}^2 )\\) \u304b\u3089 \\(C _ 2\\) \u3078\u306e \\(2\\) \u3064\u306e\u63a5\u7dda\u304c\u518d\u3073 \\(C _ 1\\) \u3068\u4ea4\u308f\u308b\u70b9\u3092 Q \\(( \\beta , {\\beta}^2 )\\) , R \\(( \\gamma , {\\gamma}^2 )\\) \u3068\u3059\u308b. \\(\\beta +\\gamma\\) \u304a\u3088\u3073 \\(\\beta \\gamma\\) \u3092 \\(\\alpha\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n

  4. (4)<\/strong>\u3000(3)<\/strong> \u306e \\(2\\) \u70b9 Q, R \u306b\u5bfe\u3057, \u76f4\u7dda QR \u306f \\(C _ 2\\) \u3068\u63a5\u3059\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

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

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

    A \\(\\left( \\sqrt{a+1} , a+1 \\right)\\) \u3068\u304a\u304f\u3053\u3068\u304c\u3067\u304d, \u76f4\u7dda OA \u306e\u5f0f\u306f\r\n\\[\r\ny = \\sqrt{a+1} x\r\n\\]\r\n\u3053\u308c\u3068, \u70b9 \\(( 0 , a )\\) \u306e\u8ddd\u96e2\u306b\u3064\u3044\u3066\u8003\u3048\u308c\u3070\r\n\\[\\begin{align}\r\n\\dfrac{| \\sqrt{a+1} \\cdot 0 -1 \\cdot a |}{\\sqrt{(a+1) +1}} & = 1 \\\\\r\na^2 & = a+2 \\\\\r\n(a-2) (a+1) & = 0 \\\\\r\n\\text{\u2234} \\quad a & = \\underline{2} \\quad ( \\ \\text{\u2235} \\ a \\geqq 0 \\ )\r\n\\end{align}\\]\r\n

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

    \\(u = \\alpha +\\beta\\) , \\(v = \\alpha \\beta\\) \u3068\u304a\u304f.
    \r\n\u76f4\u7dda PQ \u306e\u5f0f\u306f\r\n\\[\\begin{align}\r\ny & = \\dfrac{{\\alpha}^2 -{\\beta}^2}{\\alpha -\\beta} ( x -\\alpha ) +{\\alpha}^2 \\\\\r\n& = ( \\alpha +\\beta ) x -\\alpha \\beta = ux -v\r\n\\end{align}\\]\r\n\u3053\u308c\u3068, \u5186 \\(C _ 2\\) \u304c\u63a5\u3057\u3066\u3044\u308b\u306e\u3067\r\n\\[\\begin{align}\r\n\\dfrac{| -2 -v |}{\\sqrt{u^2 +1}} & = 1 \\\\\r\n(v+2)^2 & = u^2 +1 \\\\\r\n\\text{\u2234} \\quad u^2 = v^2 +4v +3\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\\begin{align}\r\n( \\alpha -\\beta )^2 -{\\alpha}^2 {\\beta}^2 & = u^2 -4v -v^2 \\\\\r\n& = 3 \\quad ... [1]\r\n\\end{align}\\]\r\n\u3067, \u4e00\u5b9a\u3068\u306a\u308b.<\/p>\r\n

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

    (2)<\/strong> \u3068\u540c\u69d8\u306e\u3053\u3068\u304c, \u76f4\u7dda PR \u306b\u3064\u3044\u3066\u3082\u6210\u7acb\u3059\u308b\u306e\u3067\r\n\\[\r\n( \\alpha -\\gamma )^2 -{\\alpha}^2 {\\gamma}^2 = 3 \\quad ... [2]\r\n\\]\r\n[1] [2] \u3088\u308a, \\(\\beta , \\gamma\\) \u306f\u65b9\u7a0b\u5f0f\r\n\\[\r\n( \\alpha -x )^2 -{\\alpha}^2 x^2 = 3\r\n\\]\r\n\u3059\u306a\u308f\u3061\r\n\\[\r\n( 1 -{\\alpha}^2 ) x^2 -2 \\alpha x +{\\alpha}^2 -3 = 0\r\n\\]\r\n\u306e\u7570\u306a\u308b \\(2\\) \u3064\u306e\u89e3\u3067\u3042\u308b.
    \r\n\u3088\u3063\u3066, \u89e3\u3068\u4fc2\u6570\u306e\u95a2\u4fc2\u3088\u308a\r\n\\[\r\n\\beta +\\gamma = \\underline{\\dfrac{2 \\alpha}{1 -{\\alpha}^2}} , \\ \\beta \\gamma = \\underline{\\dfrac{{\\alpha}^2 -3}{1 -{\\alpha}^2}}\r\n\\]\r\n

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

    \u76f4\u7dda PQ \u306e\u5f0f\u306f\r\n\\[\r\ny = ( \\beta +\\gamma ) x -\\beta \\gamma\r\n\\]\r\n\u3053\u308c\u3068, \u70b9 \\(( 0 , 2 )\\) \u3068\u306e\u8ddd\u96e2 \\(h\\) \u306f\r\n\\[\r\nh = \\dfrac{| -2 -\\beta \\gamma |}{\\sqrt{( \\beta +\\gamma )^2 +1 }} = \\dfrac{| \\beta \\gamma +2 |}{\\sqrt{( \\beta +\\gamma )^2 +1}}\r\n\\]\r\n\u3053\u3053\u3067\r\n\\[\\begin{align}\r\n\\beta \\gamma +2 & = \\dfrac{{\\alpha}^2 -3}{1 -{\\alpha}^2} +2 = -\\dfrac{1 +{\\alpha}^2}{1 -{\\alpha}^2} \\\\\r\n( \\beta +\\gamma )^2 +1 & = \\dfrac{4 {\\alpha}^2}{( 1 -{\\alpha}^2 )^2} +1 = \\left( \\dfrac{1 +{\\alpha}^2}{1 -{\\alpha}^2} \\right)^2\r\n\\end{align}\\]\r\n\u306a\u306e\u3067\r\n\\[\r\nh = 1\r\n\\]\r\n\u3088\u3063\u3066, QR \u306f\u5186 \\(C _ 2\\) \u306b\u63a5\u3059\u308b.<\/p>\r\n","protected":false},"excerpt":{"rendered":"O \u3092\u539f\u70b9\u3068\u3059\u308b\u5ea7\u6a19\u5e73\u9762\u4e0a\u306b \\[ \\text{\u653e\u7269\u7dda} \\ C _ 1 : \\ y = x^2 , \\ \\text{\u5186} \\ C _ 2 : \\ x^2 +(y-a)^2 = 1 \\quad ( a \\geqq 0 ) … \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":[120],"tags":[147,112],"class_list":["post-1199","post","type-post","status-publish","format-standard","hentry","category-waseda_r_2014","tag-waseda_r","tag-112"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1199","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=1199"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1199\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1199"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1199"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1199"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}