{"id":1978,"date":"2021-11-18T08:45:28","date_gmt":"2021-11-17T23:45:28","guid":{"rendered":"https:\/\/www.roundown.net\/nyushi\/?p=1978"},"modified":"2021-11-18T08:45:28","modified_gmt":"2021-11-17T23:45:28","slug":"tok202102","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tok202102\/","title":{"rendered":"\u6771\u5de5\u59272021\uff1a\u7b2c2\u554f"},"content":{"rendered":"<hr \/>\n<p>\\(xy\\) \u5e73\u9762\u4e0a\u306e\u6955\u5186\r\n\\[\r\nE : \\ \\dfrac{x^2}{4} +y^2 = 1\r\n\\]\r\n\u306b\u3064\u3044\u3066, \u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n<ol>\r\n<li><p><strong>(1)<\/strong>\u3000\\(a , b\\) \u3092\u5b9f\u6570\u3068\u3059\u308b.\r\n\u76f4\u7dda \\(\\ell : y = ax +b\\) \u3068\u6955\u5186 \\(E\\) \u304c\u7570\u306a\u308b \\(2\\) \u70b9\u3092\u5171\u6709\u3059\u308b\u305f\u3081\u306e \\(a , b\\) \u306e\u6761\u4ef6\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<li><p><strong>(2)<\/strong>\u3000\u5b9f\u6570 \\(a , b , c\\) \u306b\u5bfe\u3057\u3066, \u76f4\u7dda \\(\\ell : y = ax +b\\) \u3068\u76f4\u7dda \\(m : y = ax +c\\) \u304c,\r\n\u305d\u308c\u305e\u308c\u6955\u5186 \\(E\\) \u3068\u7570\u306a\u308b \\(2\\) \u70b9\u3092\u5171\u6709\u3057\u3066\u3044\u308b\u3068\u3059\u308b. \u305f\u3060\u3057, \\(b \\gt c\\) \u3068\u3059\u308b.\r\n\u76f4\u7dda \\(\\ell\\) \u3068\u6955\u5186 \\(E\\) \u306e \\(2\\) \u3064\u306e\u5171\u6709\u70b9\u306e\u3046\u3061 \\(x\\) \u5ea7\u6a19\u306e\u5c0f\u3055\u3044\u65b9\u3092 P , \u5927\u304d\u3044\u65b9\u3092 Q \u3068\u3059\u308b.\r\n\u307e\u305f, \u76f4\u7dda \\(m\\) \u3068\u6955\u5186 \\(E\\) \u306e \\(2\\) \u3064\u306e\u5171\u6709\u70b9\u306e\u3046\u3061 \\(x\\) \u5ea7\u6a19\u306e\u5c0f\u3055\u3044\u65b9\u3092 S , \u5927\u304d\u3044\u65b9\u3092 R \u3068\u3059\u308b.\r\n\u3053\u306e\u3068\u304d, \u7b49\u5f0f\r\n\\[\r\n\\overrightarrow{\\text{PQ}} = \\overrightarrow{\\text{SR}}\r\n\\]\r\n\u304c\u6210\u308a\u7acb\u3064\u305f\u3081\u306e \\(a , b , c\\) \u306e\u6761\u4ef6\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<li><p><strong>(3)<\/strong>\u3000\u6955\u5186 \\(E\\) \u4e0a\u306e \\(4\\) \u70b9\u306e\u7d44\u3067, \u305d\u308c\u3089\u3092 \\(4\\) \u9802\u70b9\u3068\u3059\u308b\u56db\u89d2\u5f62\u304c\u6b63\u65b9\u5f62\u3067\u3042\u308b\u3082\u306e\u3092\u3059\u3079\u3066\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n<hr \/>\r\n<!--more-->\r\n<h4>\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n<p><strong>(1)<\/strong><\/p>\r\n<p>\\(E\\) \u3068 \\(\\ell\\) \u306e\u5f0f\u304b\u3089 \\(y\\) \u3092\u6d88\u53bb\u3059\u308b\u3068\r\n\\[\\begin{align}\r\n\\dfrac{x^2}{4} +\\left( ax +b \\right)^2 & = 1 \\\\\r\nx^2 +4 ( a^2 x^2 +2abx +b^2 ) & = 4 \\\\\r\n\\text{\u2234} \\quad ( 4a^2 +1 ) x^2 +8abx +4 (b^2 -1 ) & = 0 \\quad ... [1]\r\n\\end{align}\\]\r\n\u3053\u308c\u304c\u7570\u306a\u308b \\(2\\) \u5b9f\u6570\u89e3\u3092\u6301\u3066\u3070\u3088\u3044\u306e\u3067, \u5224\u5225\u5f0f\u3092 \\(D\\) \u306b\u3064\u3044\u3066\r\n\\[\\begin{align}\r\n\\dfrac{D}{4} & = ( 4ab )^2 -( 4a^2 +1 ) \\cdot 4 ( b^2 -1 ) \\\\\r\n& = 16 a^2 b^2 -( 16 a^2 b^2 -16 a^2 +4b^2 -4 ) \\\\\r\n& = 4 ( 4a^2 -b^2 +1 ) \\gt 0 \\\\\r\n& \\qquad \\text{\u2234} \\quad b^2 \\lt 4a^2 +1 \\\\\r\n& \\qquad \\text{\u2234} \\quad \\underline{-\\sqrt{4a^2 +1} \\lt b \\lt \\sqrt{4a^2 +1}}\r\n\\end{align}\\]\r\n<p><strong>(2)<\/strong><\/p>\r\n[1] \u306e \\(2\\) \u89e3\u3092 \\(p , q \\ ( p\\lt q )\\) \u3068\u304a\u304f\u3068, \u3053\u308c\u306f P, Q \u306e \\(x\\) \u5ea7\u6a19\u3067\u3042\u308b.<br \/>\r\n\u540c\u69d8\u306b, S, R \u306e \\(x\\) \u5ea7\u6a19\u3092 \\(s , r \\ ( s \\lt r )\\) \u3068\u304a\u304f.<br \/>\r\n\\(\\ell\\) \u3068 \\(m\\) \u306e\u50be\u304d\u306f\u7b49\u3057\u3044\u306e\u3067, \\(\\overrightarrow{\\text{PQ}} = \\overrightarrow{\\text{SR}}\\) \u304c\u6210\u7acb\u3059\u308b\u6761\u4ef6\u306f\r\n\\[\r\nq-p = r-s\\quad ... [2]\r\n\\]\r\n[1] \u3092\u3068\u304f\u3068\r\n\\[\r\nx = \\dfrac{-8ab \\pm \\sqrt{\\dfrac{D}{4}}}{4a^2 +1}\r\n\\]\r\n\u306a\u306e\u3067\r\n\\[\r\nq-p = \\dfrac{\\sqrt{D}}{4a^2 +1} = \\dfrac{2 \\sqrt{4a^2 -b^2 +1}}{4a^2 +1}\r\n\\]\r\n\u540c\u69d8\u306b\u8003\u3048\u308c\u3070\r\n\\[\r\nr-s = \\dfrac{2 \\sqrt{4a^2 -c^2 +1}}{4a^2 +1}\r\n\\]\r\n\u306a\u306e\u3067, [2] \u306b\u4ee3\u5165\u3057\u3066\r\n\\[\\begin{align}\r\n\\dfrac{2 \\sqrt{4a^2 -b^2 +1}}{4a^2 +1} & = \\dfrac{2 \\sqrt{4a^2 -c^2 +1}}{4a^2 +1} \\\\\r\n4a^2 -b^2 +1 & = 4a^2 -c^2 +1 \\\\\r\nc^2 & = b^2 \\\\\r\n\\text{\u2234} \\quad c & = \\pm b\r\n\\end{align}\\]\r\n\\(b \\gt c\\) \u3067\u3042\u308b\u3053\u3068\u306b\u6ce8\u610f\u3059\u308c\u3070\r\n\\[\r\nc = -b \\ , \\ b \\gt 0\r\n\\]\r\n\u3088\u3063\u3066, <strong>(1)<\/strong> \u306e\u7d50\u679c\u3082\u3042\u308f\u305b\u3066, \u6c42\u3081\u308b\u6761\u4ef6\u306f\r\n\\[\r\n\\underline{c = -b \\ , \\ 0 \\lt b \\lt \\sqrt{4a^2 +1}}\r\n\\]\r\n<p><strong>(3)<\/strong><\/p>\r\n<p>P, Q, R, S \u304c\u6b63\u65b9\u5f62\u3068\u306a\u308b\u306e\u306f, \\(\\triangle \\text{OPQ}\\) \u304c\u76f4\u89d2\u4e8c\u7b49\u8fba\u4e09\u89d2\u5f62\u3068\u306a\u308b, \u3059\u306a\u308f\u3061\r\n\\[\r\n\\left\\{ \\begin{array}{ll} \\text{OP} = \\text{OQ} & ... [3] \\\\ \\angle \\text{POQ} = \\dfrac{\\pi}{2} & ... [4] \\end{array} \\right.\r\n\\]\r\n\u304c\u6210\u7acb\u3059\u308b\u3068\u304d.<\/p>\r\n<ul>\r\n<li>[3] \u306b\u3064\u3044\u3066\r\n\\[\\begin{align}\r\n\\text{OP} & = p^2 +( ap +b )^2 \\\\\r\n& = ( a^2 +1 ) p^2 +2ab p +b^2\r\n\\end{align}\\]\r\nOQ \u306b\u3064\u3044\u3066\u3082\u540c\u69d8\u306b\u8003\u3048\u3066, [3] \u306b\u4ee3\u5165\u3059\u308b\u3068\r\n\\[\\begin{align}\r\n( a^2 +1 ) p^2 +2ab p +b^2 & = ( a^2 +1 ) q^2 +2ab q +b^2 \\\\\r\n( a^2 +1 ) ( p^2 -q^2 ) +2ab ( p-q )& = 0 \\\\\r\n( p-q ) \\left\\{ ( a^2 +1 ) (p+q) +2ab \\right\\} & = 0 \\\\\r\n\\text{\u2234} \\quad p+q = -\\dfrac{2ab}{a^2 +1} & \\quad ... [5]\r\n\\end{align}\\]\r\n[1] \u306b\u3064\u3044\u3066, \u89e3\u3068\u4fc2\u6570\u306e\u95a2\u4fc2\u304b\u3089\r\n\\[\r\np+q = -\\dfrac{8ab}{4 a^2 +1} \\quad ... [6]\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, [5] [6] \u3088\u308a\r\n\\[\\begin{align}\r\n-\\dfrac{2ab}{a^2 +1} & = -\\dfrac{8ab}{4 a^2 +1} \\\\\r\n4ab ( a^2 +1 ) & = ab ( 4 a^2 +1 ) \\\\\r\n3ab & = 0 \\\\\r\n\\text{\u2234} \\quad a & = 0 \\quad ( \\ \\text{\u2235} \\ b \\gt 0 \\ ) \\quad ... [7]\r\n\\end{align}\\]<\/li>\r\n<li>[4] \u306b\u3064\u3044\u3066<br \/>\r\n[7] \u3088\u308a, \\(\\ell \\ : \\ y = b\\) \u3068\u306a\u308a, P , Q \u306f \\(y\\) \u8ef8\u306b\u3064\u3044\u3066\u5bfe\u79f0\u3068\u306a\u308b\u306e\u3067, [4] \u3068\u306a\u308b\u306e\u306f\r\n\\[\r\n\\text{P} \\ ( -b , b ) \\ , \\quad \\text{Q} \\ ( b , b )\r\n\\]\r\n\u306e\u3068\u304d.<br \/>\r\n\u3086\u3048\u306b, \\(E\\) \u306e\u5f0f\u306b\u4ee3\u5165\u3057\u3066\r\n\\[\\begin{align}\r\nb^2 +\\dfrac{b^2}{4} & = 1 \\\\\r\n\\text{\u2234} \\quad b & = \\dfrac{2}{\\sqrt{5}}\r\n\\end{align}\\]<\/li>\r\n<\/ul>\r\n<p>\u3088\u3063\u3066, <strong>(2)<\/strong> \u306e\u7d50\u679c\u3082\u3042\u308f\u305b\u308c\u3070, R \\(( b , -b )\\) , S \\(( -b , -b )\\) \u306a\u306e\u3067, \u6c42\u3081\u308b \\(4\\) \u70b9\u306e\u7d44\u306f\r\n\\[\r\n\\underline{\\left( -\\dfrac{2}{\\sqrt{5}} \\ , \\dfrac{2}{\\sqrt{5}} \\right) , \\ \\left( \\dfrac{2}{\\sqrt{5}} \\ , \\dfrac{2}{\\sqrt{5}} \\right) , \\ \\left( \\dfrac{2}{\\sqrt{5}} \\ , -\\dfrac{2}{\\sqrt{5}} \\right) , \\ \\left( -\\dfrac{2}{\\sqrt{5}} \\ , -\\dfrac{2}{\\sqrt{5}} \\right)}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(xy\\) \u5e73\u9762\u4e0a\u306e\u6955\u5186 \\[ E : \\ \\dfrac{x^2}{4} +y^2 = 1 \\] \u306b\u3064\u3044\u3066, \u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088. (1)\u3000\\(a , b\\) \u3092\u5b9f\u6570\u3068\u3059\u308b. \u76f4\u7dda \\(\\ell : y = ax +b &hellip; <a href=\"https:\/\/www.roundown.net\/nyushi\/tok202102\/\">\u7d9a\u304d\u3092\u8aad\u3080 <span class=\"meta-nav\">&rarr;<\/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":[173],"tags":[141,165],"class_list":["post-1978","post","type-post","status-publish","format-standard","hentry","category-toko_2021","tag-toko","tag-165"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1978","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=1978"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1978\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1978"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1978"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1978"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}