{"id":90,"date":"2011-11-26T23:18:26","date_gmt":"2011-11-26T14:18:26","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=90"},"modified":"2021-10-20T15:20:52","modified_gmt":"2021-10-20T06:20:52","slug":"ykr201105","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/ykr201105\/","title":{"rendered":"\u6a2a\u56fd\u5927\u7406\u7cfb2011\uff1a\u7b2c5\u554f"},"content":{"rendered":"
\n

\\(xy\\) \u5e73\u9762\u4e0a\u306b\u76f4\u7dda \\(l\\) \u304c\u3042\u308b.\r\n\u884c\u5217 \\(A = \\left( \\begin{array}{cc} a & b \\\\ c & d \\end{array} \\right)\\) \u306e\u8868\u3059 \\(1\\) \u6b21\u5909\u63db \\(f\\) \u306f, \u6b21\u306e (i) , (ii) , (iii) \u3092\u6e80\u305f\u3059.<\/p>\r\n

    \r\n

  1. (i)\u3000\u5e73\u9762\u306e\u70b9\u306e \\(f\\) \u306b\u3088\u308b\u50cf\u306f\u3059\u3079\u3066 \\(l\\) \u4e0a\u306b\u3042\u308b.<\/p><\/li>\r\n

  2. (ii)\u3000\\(f\\) \u306f \\(l\\) \u306e\u70b9\u3092\u3059\u3079\u3066\u539f\u70b9\u306b\u79fb\u3059.<\/p><\/li>\r\n

  3. (iii)\u3000\u70b9 P \u304c\u5186 \\(x^2-2x+y^2-2y+1=0\\) \u4e0a\u3092\u52d5\u304f\u3068\u304d, \\(f\\) \u306b\u3088\u308b P \u306e\u50cf\u306e \\(x\\) \u5ea7\u6a19\u306f\u6700\u5927\u5024 \\(1+\\sqrt{5}\\) , \u6700\u5c0f\u5024 \\(1-\\sqrt{5}\\) \u3092\u3068\u308b.<\/p><\/li>\r\n<\/ol>\r\n

    \u6b21\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n

      \r\n

    1. (1)<\/strong>\u3000\\(A\\) \u3092\u6c42\u3081\u3088. \u307e\u305f \\(l\\) \u306e\u65b9\u7a0b\u5f0f\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

    2. (2)<\/strong>\u3000(iii) \u3067\u6700\u5927\u5024 \\(1+\\sqrt{5}\\) \u3092\u3068\u308b\u3068\u304d\u306e P \u306e\u5ea7\u6a19\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n

      \r\n\r\n

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

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

      \u6761\u4ef6 (i) \u306b\u3064\u3044\u3066, \u70b9 \\(( x , y )\\) \u306f \\(f\\) \u306b\u3088\u3063\u3066\r\n\\[\r\n\\left( \\begin{array}{cc} a & b \\\\ c & d \\end{array} \\right) \\left( \\begin{array}{c} x \\\\ y \\end{array} \\right) = \\left( \\begin{array}{c} ax+by \\\\ cx+dy \\end{array} \\right)\r\n\\]\r\n\u306b\u79fb\u308b.
      \r\n\u3053\u308c\u304c\u76f4\u7dda \\(l : \\ px+qy+r = 0\\) \u4e0a\u306e\u70b9\u306a\u306e\u3067\r\n\\[\\begin{align}\r\np( ax+by ) +q( cx+dy ) +r & = 0 \\\\\r\n\\text{\u2234} \\quad ( ap+cq )x +( bp+dq )y +r & = 0\r\n\\end{align}\\]\r\n\u3053\u308c\u304c\u4efb\u610f\u306e \\(x , y\\) \u306b\u3064\u3044\u3066\u6210\u7acb\u3059\u308b\u306e\u3067\r\n\\[\r\n\\left\\{ \\begin{array}{ll} ap+cq =0 & ... [1] \\\\ bp+dq =0 & ... [2] \\\\ r =0 & ... [3] \\end{array} \\right.\r\n\\]\r\n[3] \u3088\u308a, \u76f4\u7dda \\(l\\) \u306f\u539f\u70b9\u3092\u901a\u308b\u76f4\u7dda\u3068\u306a\u308b.
      \r\n\u6761\u4ef6 (iii) \u306b\u3064\u3044\u3066, \u5186\u306e\u65b9\u7a0b\u5f0f\u306f\r\n\\[\r\n(x-1)^2+(y-1)^2 = 1\r\n\\]\r\n\u306a\u306e\u3067, \u70b9 P \u306f \\(( 1+\\cos \\theta , 1 +\\sin \\theta )\\) \u3068\u8868\u305b\u308b.
      \r\n\u3053\u306e\u70b9\u306f \\(f\\) \u306b\u3088\u3063\u3066,\r\n\\[\\begin{align}\r\n\\left( \\begin{array}{cc} a & b \\\\ c & d \\end{array} \\right) \\left( \\begin{array}{c} 1 +\\cos \\theta \\\\ 1 +\\sin \\theta \\end{array} \\right) & = \\left( \\begin{array}{c} a+b +a \\cos \\theta +b \\sin \\theta \\\\ c+d +c \\cos \\theta +d \\sin \\theta \\end{array} \\right) \\\\\r\n& = \\left( \\begin{array}{c} a+b +\\sqrt{a^2+b^2} \\sin ( \\theta +\\alpha ) \\\\ c+d +\\sqrt{c^2+d^2}\\sin ( \\theta +\\beta ) \\end{array} \\right)\r\n\\end{align}\\]\r\n\u306b\u79fb\u308b. \u305f\u3060\u3057\r\n\\[\r\n\\begin{array}{l} \\sin \\alpha = \\dfrac{a}{\\sqrt{a^2+b^2}} , \\ \\cos \\alpha = \\dfrac{b}{\\sqrt{a^2+b^2}} \\\\ \\sin \\beta = \\dfrac{c}{\\sqrt{c^2+d^2}} , \\ \\cos \\beta = \\dfrac{d}{\\sqrt{c^2+d^2}} \\end{array} \\quad ... [4]\r\n\\]\r\n\u3053\u306e \\(x\\) \u5ea7\u6a19\u306e\u6700\u5927\u5024, \u6700\u5c0f\u5024\u304b\u3089\r\n\\[\\begin{align}\r\na+b = 1 & , \\ \\sqrt{a^2+b^2} = \\sqrt{5} \\\\\r\n\\text{\u2234} \\quad ( a , b ) & = ( 2 , -1 ) , ( -1 , 2 )\r\n\\end{align}\\]\r\n\u6761\u4ef6 (ii) \u306b\u3064\u3044\u3066, \\(q=0\\) \u3059\u306a\u308f\u3061 \\(l : \\ x = 0\\) \u3068\u4eee\u5b9a\u3059\u308b\u3068, \\(l\\) \u4e0a\u306e\u70b9 \\(( 0 , t )\\) \u306f \\(f\\) \u306b\u3088\u3063\u3066\r\n\\[\r\n\\left( \\begin{array}{cc} a & b \\\\ c & d \\end{array} \\right) \\left( \\begin{array}{c} 0 \\\\ t \\end{array} \\right) = \\left( \\begin{array}{c} bt \\\\ dt \\end{array} \\right)\r\n\\]\r\n\u306b\u79fb\u308b.
      \r\n\\(b \\neq 0\\) \u306a\u306e\u3067, \u3053\u308c\u304c\u4efb\u610f\u306e \\(t\\) \u306b\u3064\u3044\u3066, \u539f\u70b9\u3068\u306a\u308b\u3053\u3068\u306f\u306a\u304f\u4e0d\u9069.
      \r\n\u3086\u3048\u306b\r\n\\[\r\nq \\neq 0\r\n\\]\r\n\u540c\u69d8\u306b\u8003\u3048\u308b\u3068, \\(a \\neq 0\\) \u306a\u306e\u3067, \\(p \\neq 0\\) .\r\n\u3057\u305f\u304c\u3063\u3066, \\(k=\\dfrac{p}{q}\\) \u3068\u304a\u3051\u3070, \\(l : \\ y = kx\\) \u3067, \\(l\\) \u4e0a\u306e\u70b9\u306f \\(( t , kt )\\) \u3068\u8868\u305b\u308b.<\/p>\r\n

        \r\n

      1. 1*<\/strong>\u3000\\(( a , b ) = ( 2 , -1 )\\) \u306e\u3068\u304d
        \r\n\u3053\u306e\u70b9\u306f, \\(f\\) \u306b\u3088\u3063\u3066\r\n\\[\r\n\\left( \\begin{array}{cc} 2 & -1 \\\\ c & d \\end{array} \\right) \\left( \\begin{array}{c} t \\\\ kt \\end{array} \\right) = \\left( \\begin{array}{c} ( 2-k ) t \\\\ ( c +dk ) t \\end{array} \\right)\r\n\\]\r\n\u306b\u79fb\u308b.
        \r\n\u3053\u308c\u304c\u4efb\u610f\u306e \\(t\\) \u306b\u3064\u3044\u3066\u539f\u70b9\u3068\u306a\u308b\u306e\u3067\r\n\\[\r\n\\left\\{ \\begin{array}{l} 2-k =0 \\\\ c+dk =0 \\end{array} \\right.\r\n\\]\r\n\u307e\u305f, [1] [2] \u3088\u308a\r\n\\[\\begin{gather}\r\n\\left\\{ \\begin{array}{l} 2k+c = 0 \\\\ -k+d = 0 \\end{array} \\right. \\\\\r\n\\text{\u2234} \\quad k = 2 , \\ c = -4 , \\ d = 2\r\n\\end{gather}\\]<\/li>\r\n

      2. 2*<\/strong>\u3000\\(( a , b ) = ( -1 , 2 )\\) \u306e\u3068\u304d
        \r\n\u3053\u306e\u70b9\u306f, \\(f\\) \u306b\u3088\u3063\u3066\r\n\\[\r\n\\left( \\begin{array}{cc} -1 & 2 \\\\ c & d \\end{array} \\right) \\left( \\begin{array}{c} t \\\\ kt \\end{array} \\right) = \\left( \\begin{array}{c} ( -1+2k ) t \\\\ ( c +dk ) t \\end{array} \\right)\r\n\\]\r\n\u306b\u79fb\u308b.
        \r\n\u3053\u308c\u304c\u4efb\u610f\u306e \\(t\\) \u306b\u3064\u3044\u3066\u539f\u70b9\u3068\u306a\u308b\u306e\u3067,\r\n\\[\r\n\\left\\{ \\begin{array}{l} -1+2k =0 \\\\ c+dk =0 \\end{array} \\right.\r\n\\]\r\n\u307e\u305f, [1] [2] \u3088\u308a,\r\n\\[\\begin{gather}\r\n\\left\\{ \\begin{array}{l} -k+c =0 \\\\ 2k+d =0 \\end{array} \\right. \\\\\r\n\\text{\u2234} \\quad k = \\dfrac{1}{2} , \\ c = \\dfrac{1}{2} , \\ d = -1\r\n\\end{gather}\\]<\/li>\r\n<\/ol>\r\n

        \u4ee5\u4e0a\u3088\u308a\r\n\\[\r\n\\underline{\\left\\{ \\begin{array}{lll} A = \\left( \\begin{array}{cc} 2 & -1 \\\\ -4 & 2 \\end{array} \\right) \\text{\u306e\u3068\u304d} \\quad & l : \\ y = 2x & ... [\\text{A}] \\\\ A = \\left( \\begin{array}{cc} -1 & 2 \\\\ \\dfrac{1}{2} & -1 \\end{array} \\right) \\text{\u306e\u3068\u304d} \\quad & l : \\ y = \\dfrac{1}{2} x & ... [\\text{B}] \\end{array} \\right.}\r\n\\]\r\n

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

        \u6700\u5927\u5024\u3092\u3068\u308b \\(\\theta _ M\\) \u306f\r\n\\[\r\n\\theta _ M + \\alpha = 0 \\ \\text{\u3059\u306a\u308f\u3061} \\ \\theta _ M = -\\alpha\r\n\\]\r\n

          \r\n

        1. 1*<\/strong>\u3000[A] \u306e\u3068\u304d
          \r\n[4] \u3088\u308a\r\n\\[\\begin{align}\r\n\\sin \\alpha = \\dfrac{2}{\\sqrt{5}} & , \\ \\cos \\alpha = -\\dfrac{1}{\\sqrt{5}} \\\\\r\n\\text{\u2234} \\quad \\sin \\theta _ M = -\\dfrac{2}{\\sqrt{5}} & , \\ \\cos \\theta _ M = -\\dfrac{1}{\\sqrt{5}}\r\n\\end{align}\\]\r\n\u3053\u306e\u3068\u304d\u306e P \u306e\u5ea7\u6a19\u306f\r\n\\[\r\n\\left( 1-\\dfrac{2}{\\sqrt{5}} , 1-\\dfrac{1}{\\sqrt{5}} \\right)\r\n\\]<\/li>\r\n

        2. 2*<\/strong>\u3000[B]\u306e\u3068\u304d
          \r\n[4] \u3088\u308a\r\n\\[\\begin{align}\r\n\\sin \\alpha = -\\dfrac{1}{\\sqrt{5}} & , \\cos \\alpha = \\dfrac{2}{\\sqrt{5}} \\\\\r\n\\text{\u2234} \\quad \\sin \\theta _ M = \\dfrac{1}{\\sqrt{5}} & , \\ \\cos \\theta _ M = \\dfrac{2}{\\sqrt{5}}\r\n\\end{align}\\]\r\n\u3053\u306e\u3068\u304d\u306e P \u306e\u5ea7\u6a19\u306f\r\n\\[\r\n\\left( 1+\\dfrac{1}{\\sqrt{5}} , 1+\\dfrac{2}{\\sqrt{5}} \\right)\r\n\\]<\/li>\r\n<\/ol>\r\n

          \u4ee5\u4e0a\u3088\u308a, \u6c42\u3081\u308b P \u306e\u5ea7\u6a19\u306f\r\n\\[\r\n\\underline{\\left\\{ \\begin{array}{ll} A = \\left( \\begin{array}{cc} 2 & -1 \\\\ -4 & 2 \\end{array} \\right) \\text{\u306e\u3068\u304d} \\quad & \\left( 1-\\dfrac{2}{\\sqrt{5}} , 1-\\dfrac{1}{\\sqrt{5}} \\right) \\\\ A = \\left( \\begin{array}{cc} -1 & 2 \\\\ \\dfrac{1}{2} & -1 \\end{array} \\right) \\text{\u306e\u3068\u304d} \\quad & \\left( 1+\\dfrac{1}{\\sqrt{5}} , 1+\\dfrac{2}{\\sqrt{5}} \\right) \\end{array} \\right.}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(xy\\) \u5e73\u9762\u4e0a\u306b\u76f4\u7dda \\(l\\) \u304c\u3042\u308b. \u884c\u5217 \\(A = \\left( \\begin{array}{cc} a & b \\\\ c & d \\end{array} \\right)\\) \u306e\u8868\u3059 \\(1\\) \u6b21\u5909\u63db … \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":[44],"tags":[13,9],"class_list":["post-90","post","type-post","status-publish","format-standard","hentry","category-yokokoku_r_2011","tag-13","tag-yokokoku_r"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/90","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=90"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/90\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=90"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=90"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=90"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}