{"id":1893,"date":"2021-09-22T21:09:42","date_gmt":"2021-09-22T12:09:42","guid":{"rendered":"https:\/\/www.roundown.net\/nyushi\/?p=1893"},"modified":"2021-09-22T21:11:17","modified_gmt":"2021-09-22T12:11:17","slug":"tbr201601","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tbr201601\/","title":{"rendered":"\u7b51\u6ce2\u5927\u7406\u7cfb2016\uff1a\u7b2c1\u554f"},"content":{"rendered":"
\n

\\(k\\) \u3092\u5b9f\u6570\u3068\u3059\u308b. \\(xy\\) \u5e73\u9762\u306e\u66f2\u7dda \\(C _ 1 : \\ y = x^2\\) \u3068 \\(C _ 2 : \\ y = -x^2 +2kx +1 -k^2\\) \u304c\u7570\u306a\u308b\u5171\u6709\u70b9 P , Q \u3092\u6301\u3064\u3068\u3059\u308b. \u305f\u3060\u3057\u70b9 P , Q \u306e \\(x\\) \u5ea7\u6a19\u306f\u6b63\u3067\u3042\u308b\u3068\u3059\u308b. \u307e\u305f, \u539f\u70b9\u3092 O \u3068\u3059\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(k\\) \u306e\u3068\u308a\u3046\u308b\u5024\u306e\u7bc4\u56f2\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(k\\) \u304c (1)<\/strong> \u306e\u7bc4\u56f2\u3092\u52d5\u304f\u3068\u304d, \\(\\triangle \\text{OPQ}\\) \u306e\u91cd\u5fc3 G \u306e\u8ecc\u8de1\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\\(\\triangle \\text{OPQ}\\) \u306e\u9762\u7a4d\u3092 \\(S\\) \u3068\u3059\u308b\u3068\u304d, \\(S^2\\) \u3092 \\(k\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n

  4. (4)<\/strong>\u3000\\(k\\) \u304c (1)<\/strong> \u306e\u7bc4\u56f2\u3092\u52d5\u304f\u3068\u3059\u308b. \\(\\triangle \\text{OPQ}\\) \u306e\u9762\u7a4d\u304c\u6700\u5927\u3068\u306a\u308b\u3088\u3046\u306a \\(k\\) \u306e\u5024\u3068, \u305d\u306e\u3068\u304d\u306e\u91cd\u5fc3 G \u306e\u5ea7\u6a19\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

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

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

    \\(C_1 , C_2\\) \u306e\u5f0f\u3088\u308a\r\n\\[\\begin{align}\r\nx^2 = -x^2 +2kx +1 & -k^2 \\\\\r\n\\text{\u2234} \\quad 2x^2 -2kx +k^2 -1 & = 0 \\quad ... [1] \\ .\r\n\\end{align}\\]\r\n[1] \u306e \\(2\\) \u3064\u306e\u89e3\u304c P , Q \u306e \\(x\\) \u5ea7\u6a19\u306a\u306e\u3067, \u5224\u5225\u5f0f \\(D\\) \u306b\u3064\u3044\u3066\r\n\\[\\begin{align}\r\n\\dfrac{D}{4} & = k^2 -2 ( k^2 -1 ) \\\\\r\n& = 2 -k^2 \\gt 0 \\\\\r\n\\text{\u2234} \\quad -\\sqrt{2} & \\lt k \\lt \\sqrt{2} \\quad ... [2] \\ .\r\n\\end{align}\\]\r\n\u307e\u305f, \u89e3\u3068\u4fc2\u6570\u306e\u95a2\u4fc2\u304b\u3089\r\n\\[\\begin{align}\r\n\\dfrac{2k}{2} \\gt 0 \\ & , \\ \\dfrac{k^2 -1}{2} \\gt 0 \\\\\r\n\\text{\u2234} \\quad k & \\gt 1 \\quad ... [3] \\ .\r\n\\end{align}\\]\r\n[2] [3] \u3088\u308a, \u6c42\u3081\u308b\u7bc4\u56f2\u306f\r\n\\[\r\n\\underline{1 \\lt k \\lt \\sqrt{2}} \\ .\r\n\\]\r\n

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

    P , Q \u306e \\(x\\) \u5ea7\u6a19\u3092 \\(p , q \\ ( p \\gt q )\\) \u3068\u304a\u304f\u3068\r\n\\[\r\np+q = k \\ , \\ pq = \\dfrac{k^2 -1}{2} \\quad ... [4] \\ .\r\n\\]\r\nP \\(( p , p^2 )\\) , Q \\(( q , q^2 )\\) \u306a\u306e\u3067\r\n\\[\r\n\\text{G} \\left( \\dfrac{p+q}{3} , \\dfrac{p^2 +q^2}{3} \\right) \\ .\r\n\\]\r\n[4] \u3092\u7528\u3044\u3066\r\n\\[\\begin{align}\r\np^2 +q^2 & = ( p+q )^2 -2pq \\\\\r\n& = k^2 -2 \\cdot \\dfrac{k^2 -1}{2} = 1 \\ .\r\n\\end{align}\\]\r\n\u3086\u3048\u306b, G \\(\\left( \\dfrac{k}{3} , 1 \\right)\\) .
    \r\n(1)<\/strong> \u306e\u7d50\u679c\u3088\u308a, \\(\\dfrac{1}{3} \\lt \\dfrac{k}{3} \\lt \\dfrac{\\sqrt{2}}{3}\\) \u306a\u306e\u3067, \u6c42\u3081\u308b\u8ecc\u8de1\u306f\r\n\\[\r\n\\underline{\\text{\u7dda\u5206} \\ : \\ y = \\dfrac{1}{3} \\ \\left( \\dfrac{1}{3} \\lt x \\lt \\dfrac{\\sqrt{2}}{3} \\right)} \\ .\r\n\\]\r\n

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

    \\[\\begin{align}\r\nS & = \\dfrac{1}{2} \\left| p q^2 -p^2 q \\right| \\\\\r\n& = \\dfrac{1}{2} pq | p-q | \\ .\r\n\\end{align}\\]\r\n\u306a\u306e\u3067, [4] \u3082\u7528\u3044\u3066\r\n\\[\\begin{align}\r\nS^2 & = \\dfrac{1}{4} (pq)^2 \\left\\{ (p+q)^2 -4pq \\right\\} \\\\\r\n& = \\dfrac{1}{4} \\cdot \\dfrac{( k^2 -1 )^2}{4} \\left\\{ k^2 -2 ( k^2 -1 ) \\right\\} \\\\\r\n& = \\underline{\\dfrac{1}{16} ( k^2 -1 )^2 ( 2 -k^2 )} \\ .\r\n\\end{align}\\]\r\n

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

    \\(t =k^2\\) \u3068\u304a\u304f\u3068, \\(1 \\lt t \\lt 2\\) ... [5] .
    \r\n\\(f(t) = ( t-1 )^2 ( 2-t )\\) \u3068\u304a\u3051\u3070, [5] \u306b\u304a\u3044\u3066 \\(f(t)\\) \u304c\u6700\u5927\u306b\u306a\u308b\u3068\u304d, \\(S\\) \u3082\u6700\u5927\u3068\u306a\u308b.\r\n\\[\\begin{align}\r\nf'(t) & = 2 ( t-1 ) ( 2-t ) -( t-1 )^2 \\\\\r\n& = ( t-1 ) ( 5 -3t ) \\ .\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, [5] \u306b\u304a\u3051\u308b \\(f(t)\\) \u306e\u5897\u6e1b\u306f\u4e0b\u8868\u306e\u901a\u308a.\r\n\\[\r\n\\begin{array}{c|ccccc} t & ( 1 ) & \\cdots & \\dfrac{5}{3} & \\cdots & ( 2 ) \\\\ \\hline f'(t) & (0) & + & 0 & - & \\\\ \\hline f(t) & & \\nearrow & \\text{\u6700\u5927} & \\searrow & \\end{array}\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(f(t)\\) \u306e\u6700\u5927\u5024\u306f\r\n\\[\r\nf \\left( \\dfrac{5}{3} \\right) = \\left( \\dfrac{2}{3} \\right)^2 \\cdot \\dfrac{1}{3} = \\dfrac{4}{27} \\ .\r\n\\]\r\n\u3088\u3063\u3066, \\(S\\) \u304c\u6700\u5927\u3068\u306a\u308b\u306e\u306f\r\n\\[\r\nk = \\underline{\\dfrac{\\sqrt{15}}{3}}\r\n\\]\r\n\u306e\u3068\u304d\u3067, \u3053\u306e\u3068\u304d, G \u306e\u5ea7\u6a19\u306f\r\n\\[\r\n\\underline{\\left(\\dfrac{\\sqrt{15}}{9} , \\dfrac{1}{3} \\right)} \\ .\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(k\\) \u3092\u5b9f\u6570\u3068\u3059\u308b. \\(xy\\) \u5e73\u9762\u306e\u66f2\u7dda \\(C _ 1 : \\ y = x^2\\) \u3068 \\(C _ 2 : \\ y = -x^2 +2kx +1 -k^2\\) \u304c\u7570\u306a\u308b\u5171\u6709\u70b9 P , Q \u3092\u6301\u3064\u3068\u3059\u308b. \u305f … \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":[158],"tags":[144,162],"class_list":["post-1893","post","type-post","status-publish","format-standard","hentry","category-tsukuba_r_2016","tag-tsukuba_r","tag-162"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1893","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=1893"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1893\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1893"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1893"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1893"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}