{"id":1188,"date":"2015-07-25T21:32:13","date_gmt":"2015-07-25T12:32:13","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=1188"},"modified":"2021-10-20T13:56:40","modified_gmt":"2021-10-20T04:56:40","slug":"ykr201403","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/ykr201403\/","title":{"rendered":"\u6a2a\u56fd\u5927\u7406\u7cfb2014\uff1a\u7b2c3\u554f"},"content":{"rendered":"
\n

O \u3092\u539f\u70b9\u3068\u3059\u308b\u5ea7\u6a19\u7a7a\u9593\u306b, \\(4\\) \u70b9\r\n\\[\r\n\\text{A} \\ ( -2 , 1 , 3 ) , \\ \\text{B} \\ ( s , 3 , -1 ) , \\ \\text{C} \\ ( 1 , 3 , 4 ) , \\ \\text{D} \\ ( t , 2t , 2t )\r\n\\]\r\n\u304c\u3042\u308b. \u305f\u3060\u3057, \\(s , t\\) \u306f\u5b9f\u6570\u3067 \\(t \\neq 0\\) \u3067\u3042\u308b. A \u3092\u901a\u308a \\(\\overrightarrow{\\text{OC}}\\) \u306b\u5e73\u884c\u306a\u76f4\u7dda\u3068, B \u3092\u901a\u308a \\(\\overrightarrow{\\text{OD}}\\) \u306b\u5e73\u884c\u306a\u76f4\u7dda\u304c\u70b9 P \u3067\u4ea4\u308f\u308b\u3068\u3059\u308b. \u6b21\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n

    \r\n
  1. (1)<\/strong>\u3000\\(s\\) \u306e\u5024\u304a\u3088\u3073 P \u306e\u5ea7\u6a19\u3092\u6c42\u3081\u3088.<\/li>\r\n<\/ol>\r\n

    \u3000\u4ee5\u4e0b\u3067\u306f, \\(\\triangle \\text{PAB} \\sim \\triangle \\text{OCD}\\) \u3092\u4eee\u5b9a\u3059\u308b.<\/p>\r\n

      \r\n

    1. (2)<\/strong>\u3000\\(t\\) \u306e\u5024\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

    2. (3)<\/strong>\u3000D \u304b\u3089\u5e73\u9762 PAB \u306b\u4e0b\u308d\u3057\u305f\u5782\u7dda\u3092 DH \u3068\u3059\u308b\u3068\u304d, H \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

      A \u3092\u901a\u308a \\(\\overrightarrow{\\text{OC}}\\) \u306b\u5e73\u884c\u306a\u76f4\u7dda\u3092 \\(m\\) , B \u3092\u901a\u308a \\(\\overrightarrow{\\text{OD}}\\) \u306b\u5e73\u884c\u306a\u76f4\u7dda\u3092 \\(\\ell\\) \u3068\u304a\u304f.
      \r\n\\[\\begin{align}\r\nm : \\ \\left( \\begin{array}{c} x \\\\ y \\\\ z \\end{array} \\right) & = \\left( \\begin{array}{c} -2 \\\\ 1 \\\\ 3 \\end{array} \\right) +u \\left( \\begin{array}{c} 1 \\\\ 3 \\\\ 4 \\end{array} \\right) = \\left( \\begin{array}{c} u-2 \\\\ 3u-1 \\\\ 4y=3 \\end{array} \\right) \\\\\r\n\\ell : \\ \\left( \\begin{array}{c} x \\\\ y \\\\ z \\end{array} \\right) & = \\left( \\begin{array}{c} s \\\\ 3 \\\\ -1 \\end{array} \\right) +v \\left( \\begin{array}{c} t \\\\ 2t \\\\ 2t \\end{array} \\right) = \\left( \\begin{array}{c} tv +s \\\\ 2tv +3 \\\\ 2tv -1 \\end{array} \\right)\r\n\\end{align}\\]\r\nP \u306f \\(m\\) \u3068 \\(\\ell\\) \u306e\u4ea4\u70b9\u306a\u306e\u3067\r\n\\[\r\n\\left\\{ \\begin{array}{ll} u-2 = tv +s & ... [1] \\\\ 3u+1 = 2tv +3 & ... [2] \\\\ 4u+3 = 2tv -1 & ... [3] \\end{array} \\right.\r\n\\]\r\n\\([3] -[2]\\) \u3088\u308a\r\n\\[\\begin{align}\r\nu+2 & = -4 \\\\\r\n\\text{\u2234} \\quad u & = -6\r\n\\end{align}\\]\r\n[3] \u3088\u308a\r\n\\[\r\nv = \\dfrac{4 (u+1)}{2t} = -\\dfrac{10}{t}\r\n\\]\r\n[1] \u306b\u4ee3\u5165\u3057\u3066\r\n\\[\r\ns = u-2 -tv = -6 -2 +10 = \\underline{2}\r\n\\]\r\n\u3053\u306e\u3068\u304d\r\n\\[\r\n\\text{P} \\ \\underline{( -8 , -17 , -21 )}\r\n\\]\r\n

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

      \\(\\angle \\text{APB} = \\angle \\text{COD}\\) \u306a\u306e\u3067, \\(\\triangle \\text{PAB} \\sim \\triangle \\text{OCD}\\) \u3068\u306a\u308b\u6761\u4ef6\u306f\r\n\\[\r\n\\text{PA} : \\text{PB} = \\text{OC} : \\text{OD} \\quad ... [4]\r\n\\]\r\n\u3053\u3053\u3067\r\n\\[\\begin{align}\r\n\\text{PA} & = \\sqrt{6^2 +18^2 +24^2} \\\\\r\n& = 6 \\sqrt{1+9+4} = 6 \\sqrt{14} , \\\\\r\n\\text{PB} & = \\sqrt{10^2 +20^2 +20^2} \\\\\r\n& = 10 \\sqrt{1+4+4} = 30 , \\\\\r\n\\text{OC} & = \\sqrt{1^2 +3^2 +3^2} = \\sqrt{14} , \\\\\r\n\\text{OD} & = \\sqrt{t^2 +(2t)^2 +(2t)^2} = 3t\r\n\\end{align}\\]\r\n\u306a\u306e\u3067, [4] \u3088\u308a\r\n\\[\\begin{align}\r\n6 \\sqrt{14} : 30 & = \\sqrt{14} : 3t \\\\\r\n30 \\sqrt{14} & = 18 t \\sqrt{14} \\\\\r\n\\text{\u2234} \\quad t & = \\underline{\\dfrac{5}{3}}\r\n\\end{align}\\]\r\n

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

      \\(\\overrightarrow{\\text{PA}} \\, \/ \\hspace{-0.2em} \/ \\left( \\begin{array}{c} 1 \\\\ 3 \\\\ 4 \\end{array} \\right)\\) , \\(\\overrightarrow{\\text{PB}} \\, \/ \\hspace{-0.2em} \/ \\left( \\begin{array}{c} 1 \\\\ 2 \\\\ 2 \\end{array} \\right)\\) \u3092\u5229\u7528\u3059\u308c\u3070, H \u306f\u5e73\u9762 PAB \u4e0a\u306b\u3042\u308b\u306e\u3067, \u5b9f\u6570 \\(\\alpha , \\beta\\) \u3092\u7528\u3044\u3066\r\n\\[\\begin{align}\r\n\\overrightarrow{\\text{OH}} & = \\left( \\begin{array}{c} -8 \\\\ -17 \\\\ -21 \\end{array} \\right) +\\alpha \\left( \\begin{array}{c} 1 \\\\ 3 \\\\ 4 \\end{array} \\right) +\\beta \\left( \\begin{array}{c} 1 \\\\ 2 \\\\ 2 \\end{array} \\right) \\\\\r\n& = \\left( \\begin{array}{c} -8 +\\alpha +\\beta \\\\ -17 +3 \\alpha +2 \\beta \\\\ -21 +4 \\alpha +2 \\beta \\end{array} \\right)\r\n\\end{align}\\]\r\n\u3068\u8868\u305b\u308b.\r\n\\[\\begin{align}\r\n\\overrightarrow{\\text{DH}} & = \\overrightarrow{\\text{OH}} -\\overrightarrow{\\text{OD}} \\\\\r\n& = \\left( \\begin{array}{c} -\\frac{29}{3} +\\alpha +\\beta \\\\ -\\frac{61}{3} +3 \\alpha +2 \\beta \\\\ -\\frac{73}{3} +4 \\alpha +2 \\beta \\end{array} \\right)\r\n\\end{align}\\]\r\n\\(\\overrightarrow{\\text{DH}} \\perp \\overrightarrow{\\text{OC}}\\) \u3088\u308a\r\n\\[\\begin{align}\r\n\\overrightarrow{\\text{DH}} \\cdot \\overrightarrow{\\text{OC}} & = \\left( -\\dfrac{29}{3} +\\alpha +\\beta \\right) +3 \\left( -\\dfrac{61}{3} +3 \\alpha +2 \\beta \\right) \\\\\r\n& \\qquad +4 \\left( -\\dfrac{73}{3} +4 \\alpha +2 \\beta \\right) \\\\\r\n& = -168 +26 \\alpha +15 \\beta = 0 \\quad ... [5]\r\n\\end{align}\\]\r\n\\(\\overrightarrow{\\text{DH}} \\perp \\overrightarrow{\\text{OD}}\\) \u3088\u308a\r\n\\[\\begin{align}\r\n\\overrightarrow{\\text{DH}} \\cdot \\left( \\begin{array}{c} 1 \\\\ 2 \\\\ 2 \\end{array} \\right) & = \\left( -\\dfrac{29}{3} +\\alpha +\\beta \\right) +2 \\left( -\\dfrac{61}{3} +3 \\alpha +2 \\beta \\right) \\\\\r\n& \\qquad +2 \\left( -\\dfrac{73}{3} +4 \\alpha +2 \\beta \\right) \\\\\r\n& = -99 +15 \\alpha +9 \\beta = 0 \\\\\r\n\\text{\u2234} \\quad & -33 +5 \\alpha +3 \\beta = 0 \\quad ... [6]\r\n\\end{align}\\]\r\n\\([6] \\times 4 -[5]\\) \u3088\u308a\r\n\\[\\begin{align}\r\n-3 +\\alpha & = 0 \\\\\r\n\\text{\u2234} \\quad \\alpha & = 3\r\n\\end{align}\\]\r\n[6] \u3088\u308a\r\n\\[\r\n\\beta = \\dfrac{33 -15}{5} = 6\r\n\\]\r\n\u3088\u3063\u3066\r\n\\[\r\n\\overrightarrow{\\text{OH}} = \\left( \\begin{array}{c} -8 +3 +6 \\\\ -17 +9 +12 \\\\ -21 +12 +12 \\end{array} \\right) = \\left( \\begin{array}{c} 1 \\\\ 4 \\\\ 3 \\end{array} \\right)\r\n\\]\r\n\u3059\u306a\u308f\u3061\r\n\\[\r\n\\text{H} \\ \\underline{( 1 , 4 , 3 )}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"O \u3092\u539f\u70b9\u3068\u3059\u308b\u5ea7\u6a19\u7a7a\u9593\u306b, \\(4\\) \u70b9 \\[ \\text{A} \\ ( -2 , 1 , 3 ) , \\ \\text{B} \\ ( s , 3 , -1 ) , \\ \\text{C} \\ ( 1 , 3 , 4 ) … \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":[122],"tags":[112,9],"class_list":["post-1188","post","type-post","status-publish","format-standard","hentry","category-yokokoku_r_2014","tag-112","tag-yokokoku_r"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1188","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=1188"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1188\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1188"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1188"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1188"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}