\u5e73\u9762\u4e0a\u306e \\(2\\) \u3064\u306e\u30d9\u30af\u30c8\u30eb \\(\\overrightarrow{a}\\) \u3068 \\(\\overrightarrow{b}\\) \u306f\u96f6\u30d9\u30af\u30c8\u30eb\u3067\u306f\u306a\u304f, \\(\\overrightarrow{a}\\) \u3068 \\(\\overrightarrow{b}\\) \u306e\u306a\u3059\u89d2\u5ea6\u306f \\(60^{\\circ}\\) \u3067\u3042\u308b.\r\n\u3053\u306e\u3068\u304d\r\n\\[\r\nr = \\dfrac{\\left| \\overrightarrow{a} +2 \\overrightarrow{b} \\right|}{\\left| 2 \\overrightarrow{a} +\\overrightarrow{b} \\right|}\r\n\\]\r\n\u306e\u3068\u308a\u3046\u308b\u5024\u306e\u7bc4\u56f2\u3092\u6c42\u3081\u3088.<\/p>\r\n
\\(a = \\left| \\overrightarrow{a} \\right|\\) , \\(b = \\left| \\overrightarrow{b} \\right|\\) \u3068\u304a\u304f\u3068, \\(a \\gt 0\\) , \\(b \\gt 0\\) .
\r\n\\[\r\n\\overrightarrow{a} \\cdot \\overrightarrow{b} = ab \\cos 60^{\\circ} = \\dfrac{1}{2} ab\r\n\\]\r\n\u3092\u7528\u3044\u308c\u3070\r\n\\[\\begin{align}\r\n\\left| \\overrightarrow{a} +2 \\overrightarrow{b} \\right|^2 & = a^2 +2ab +4 b^2 \\ , \\\\\r\n\\left| 2 \\overrightarrow{a} +\\overrightarrow{b} \\right|^2 & = 4 a^2 +2ab +b^2\r\n\\end{align}\\]\r\n\\(t = \\dfrac{a}{b} ( \\gt 0 )\\) \u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\nr^2 & = \\dfrac{a^2 +2ab +4 b^2}{4 a^2 +2ab +b^2} \\\\\r\n& = \\dfrac{t^2 +2t +4}{4t^2 +2t +1}\r\n\\end{align}\\]\r\n\u3053\u308c\u3092\u5909\u5f62\u3057\u3066\r\n\\[\\begin{gather}\r\n( 4t^2 +2t +1 ) r^2 = t^2 +2t +4 \\\\\r\n\\text{\u2234} \\quad ( 4r^2 -1 ) t^2 +2 ( r^2 -1 ) t +r^2 -4 = 0 \\quad ... [1]\r\n\\end{gather}\\]\r\n\u3057\u305f\u304c\u3063\u3066, [1] \u304c \\(t \\gt 0\\) \u306b\u89e3\u3092\u3082\u3064\u6761\u4ef6\u3092\u6c42\u3081\u308c\u3070\u3088\u3044.<\/p>\r\n
1*<\/strong>\u3000\\(r = \\dfrac{1}{2}\\) \u306e\u3068\u304d 2*<\/strong>\u3000\\(r \\neq \\dfrac{1}{2}\\) \u306e\u3068\u304d \u4ee5\u4e0a\u3088\u308a, \u6c42\u3081\u308b\u5024\u306e\u7bc4\u56f2\u306f\r\n\\[\r\n\\underline{\\dfrac{1}{2} \\lt r \\lt 2}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u5e73\u9762\u4e0a\u306e \\(2\\) \u3064\u306e\u30d9\u30af\u30c8\u30eb \\(\\overrightarrow{a}\\) \u3068 \\(\\overrightarrow{b}\\) \u306f\u96f6\u30d9\u30af\u30c8\u30eb\u3067\u306f\u306a\u304f, \\(\\overrightarrow{a}\\) \u3068 \\(\\over … \u7d9a\u304d\u3092\u8aad\u3080
\r\n[1] \u3088\u308a\r\n\\[\\begin{align}\r\n-2 \\cdot \\dfrac{3}{4} t -\\dfrac{15}{4} & = 0 \\\\\r\n\\text{\u2234} \\quad t = -\\dfrac{5}{2}\r\n\\end{align}\\]\r\n\u3053\u308c\u306f, \\(t \\gt 0\\) \u306b\u89e3\u3092\u3082\u305f\u306a\u3044\u306e\u3067\u4e0d\u9069.<\/p><\/li>\r\n
\r\n\\(4 r^2 -1 \\gt 2 ( r^2 -1 ) \\gt r^2 -4 \\) \u3067\u3042\u308b\u3053\u3068\u306b\u6ce8\u610f\u3059\u308c\u3070, \u6b63\u306e\u5b9f\u6570\u89e3\u3092\u3082\u3064\u6761\u4ef6\u306f\r\n\\[\\begin{gather}\r\n4 r^2 -1 \\gt 0 \\ \\text{\u304b\u3064} \\ r^2 -4 \\lt 0 \\\\\r\n\\text{\u2234} \\quad \\dfrac{1}{2} \\lt r \\lt 2\r\n\\end{gather}\\]<\/li>\r\n<\/ol>\r\n