{"id":1033,"date":"2015-04-10T23:26:31","date_gmt":"2015-04-10T14:26:31","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=1033"},"modified":"2021-10-31T06:37:31","modified_gmt":"2021-10-30T21:37:31","slug":"tkb201303","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tkb201303\/","title":{"rendered":"\u6771\u5927\u6587\u7cfb2013\uff1a\u7b2c3\u554f"},"content":{"rendered":"
\n

\\(a , b\\) \u3092\u5b9f\u6570\u306e\u5b9a\u6570\u3068\u3059\u308b. \u5b9f\u6570 \\(x , y\\) \u304c\r\n\\[\r\nx^2 +y^2 \\leqq 25 , \\ 2x+y \\leqq 5\r\n\\]\r\n\u3092\u3068\u3082\u306b\u307f\u305f\u3059\u3068\u304d, \\(z = x^2+y^2-2ax-2by\\) \u306e\u6700\u5c0f\u5024\u3092\u6c42\u3081\u3088.<\/p>\r\n

\r\n\r\n

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

\u4e0e\u3048\u3089\u308c\u305f\u4e0d\u7b49\u5f0f\u3092\u307f\u305f\u3059\u9818\u57df \\(D\\) \u306f, \u4e0b\u56f3\u659c\u7dda\u90e8\uff08\u5883\u754c\u542b\u3080\uff09.<\/p>\r\n\"tokyob20130301\"\r\n

\u5b9a\u70b9 A \\(( a , b )\\) \u306b\u5bfe\u3057\u3066, \\(D\\) \u306b\u542b\u307e\u308c\u308b\u70b9 P \\(( x , y )\\) \u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\nz & = (x-a)^2 +(y-b)^2 -\\left( a^2 +b^2 \\right) \\\\\r\n& = \\text{AP}^2 -\\text{OA}^2\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, AP \u304c\u6700\u5c0f\u3068\u306a\u308b\u3068\u304d\u306b \\(z\\) \u3082\u6700\u5c0f\u306b\u306a\u308b.
\r\nAP \u306e\u6700\u5c0f\u5024\u3092 \\(m\\) \u3068\u304a\u304f.
\r\n\\(D\\) \u306e\u5f62\u72b6\u306b\u6ce8\u610f\u3057\u3066, \u70b9A\u306e\u4f4d\u7f6e\u306b\u3088\u3063\u3066\u5834\u5408\u5206\u3051\u3057\u3066\u8003\u3048\u308b.<\/p>\r\n\"tokyob20130302\"\r\n

    \r\n

  1. (A)<\/strong>\u3000\\(a^2 +b^2 \\leqq 25\\) , \\(2a+b \\leqq 5\\) \u306e\u3068\u304d\r\n\\[\r\nm = 0\r\n\\]\r\n\u3086\u3048\u306b, \\(z\\) \u306e\u6700\u5c0f\u5024\u306f\r\n\\[\r\nm^2 -\\left( a^2 +b^2 \\right) = -\\left( a^2 +b^2 \\right)\r\n\\]<\/li>\r\n

  2. (B)<\/strong>\u3000\\(a^2 +b^2 \\gt 25\\) \u304b\u3064 \u300c \\(a \\leqq 0\\) \u307e\u305f\u306f \\(b \\leqq -\\dfrac{3a}{4}\\) \u300d\u306e\u3068\u304d\r\n\\[\r\nm = \\sqrt{a^2 +b^2} -5\r\n\\]\r\n\u3086\u3048\u306b, \\(z\\) \u306e\u6700\u5c0f\u5024\u306f\r\n\\[\r\nm^2 -\\left( a^2 +b^2 \\right) = 25 -10 \\sqrt{a^2 +b^2}\r\n\\]<\/li>\r\n

  3. (C)<\/strong>\u3000\\(2a+b \\gt 5\\) , \\(\\dfrac{a}{2} -5 \\leqq b \\leqq \\dfrac{a}{2} +5\\) \u306e\u3068\u304d\r\n\\[\r\nm = \\dfrac{| 2a+b-5 |}{\\sqrt{2^2 +1^2}} = \\dfrac{( 2a+b-5 )}{\\sqrt{5}}\r\n\\]\r\n\u3086\u3048\u306b, \\(z\\) \u306e\u6700\u5c0f\u5024\u306f\r\n\\[\\begin{align}\r\nm^2 -\\left( a^2 +b^2 \\right) & = \\dfrac{( 2a+b-5 )^2}{5} -\\left( a^2 +b^2 \\right) \\\\\r\n& = 5 +4a -2b -\\dfrac{(a-2b)^2}{5}\r\n\\end{align}\\]<\/li>\r\n

  4. (D)<\/strong>\u3000\\(a \\gt 0\\) , \\(b \\gt \\dfrac{a}{2} +5\\) \u306e\u3068\u304d\r\n\\[\r\nm = \\sqrt{a^2 +(b-5)^2}\r\n\\]\r\n\u3086\u3048\u306b, \\(z\\) \u306e\u6700\u5c0f\u5024\u306f\r\n\\[\r\nm^2 -\\left( a^2 +b^2 \\right) = 25 -10b\r\n\\]<\/li>\r\n

  5. (E)<\/strong>\u3000\\(-\\dfrac{3a}{4} \\lt b \\lt \\dfrac{a}{2} -5\\) \u306e\u3068\u304d\r\n\\[\r\nm = \\sqrt{(a-4)^2 +(b+3)^2}\r\n\\]\r\n\u3086\u3048\u306b, \\(z\\) \u306e\u6700\u5c0f\u5024\u306f\r\n\\[\r\nm^2 -\\left( a^2 +b^2 \\right) = 25 -4a +6b\r\n\\]<\/li>\r\n<\/ol>\r\n

    \u4ee5\u4e0a\u3088\u308a, \\(z\\) \u306e\u6700\u5c0f\u5024\u306f\r\n\\[\r\n\\underline{\\left\\{ \\begin{array}{ll} -\\left( a^2 +b^2 \\right) & \\left( \\ x^2 +y^2 \\leqq 25 , \\ 2x+y \\leqq 5 \\text{\u306e\u3068\u304d} \\right) \\\\ 25 -10 \\sqrt{a^2 +b^2} & \\left( \\ a^2 +b^2 \\gt 25 \\text{\u304b\u3064 \u300c} a \\leqq 0 \\text{\u307e\u305f\u306f} b \\leqq -\\dfrac{3a}{4} \\text{\u300d \u306e\u3068\u304d} \\right) \\\\ 5 +4a -2b -\\dfrac{(a-2b)^2}{5} & \\left( \\ 2a+b \\gt 5 , \\ \\dfrac{a}{2} -5 \\leqq b \\leqq \\dfrac{a}{2} +5 \\text{\u306e\u3068\u304d} \\right) \\\\ 25 -10b & \\left( \\ a \\gt 0 , \\ b \\gt \\dfrac{a}{2} +5 \\text{\u306e\u3068\u304d} \\right) \\\\ 25 -4a +6b & \\left( \\ -\\dfrac{3a}{4} \\lt b \\lt \\dfrac{a}{2} -5 \\text{\u306e\u3068\u304d} \\right) \\end{array} \\right.}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(a , b\\) \u3092\u5b9f\u6570\u306e\u5b9a\u6570\u3068\u3059\u308b. \u5b9f\u6570 \\(x , y\\) \u304c \\[ x^2 +y^2 \\leqq 25 , \\ 2x+y \\leqq 5 \\] \u3092\u3068\u3082\u306b\u307f\u305f\u3059\u3068\u304d, \\(z = x^2+y^2-2ax-2by\\ … \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":[106],"tags":[149,111],"class_list":["post-1033","post","type-post","status-publish","format-standard","hentry","category-tokyo_b_2013","tag-tokyo_b","tag-111"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1033","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=1033"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1033\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1033"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1033"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1033"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}