\\(0 \\lt \\theta \\lt \\dfrac{\\pi}{2}\\) \u3068\u3059\u308b. \\(2\\) \u3064\u306e\u66f2\u7dda\r\n\\[\r\nC _ 1 : \\ x^2 + 3y^2 = 3 , \\quad C _ 2 : \\ \\dfrac{x^2}{\\cos^2 \\theta} - \\dfrac{y^2}{\\sin^2 \\theta} = 2\r\n\\]\r\n\u306e\u4ea4\u70b9\u306e\u3046\u3061, \\(x\\) \u5ea7\u6a19\u3068 \\(y\\) \u5ea7\u6a19\u304c\u3068\u3082\u306b\u6b63\u3067\u3042\u308b\u3082\u306e\u3092 P \u3068\u3059\u308b.\r\nP \u306b\u304a\u3051\u308b \\(C _ 1 , C _ 2\\) \u306e\u63a5\u7dda\u3092\u305d\u308c\u305e\u308c \\(\\ell _ 1 , \\ell _ 2\\) \u3068\u3057, \\(y\\) \u8ef8\u3068 \\(\\ell _ 1 , \\ell _ 2\\) \u306e\u4ea4\u70b9\u3092\u305d\u308c\u305e\u308c Q , R \u3068\u3059\u308b.\r\n\\(\\theta\\) \u304c \\(0 \\lt \\theta \\lt \\dfrac{\\pi}{2}\\) \u306e\u7bc4\u56f2\u3092\u52d5\u304f\u3068\u304d, \u7dda\u5206 QR \u306e\u9577\u3055\u306e\u6700\u5c0f\u5024\u3092\u6c42\u3081\u3088.<\/p>\r\n
P \\(( p , q ) \\ ( p \\gt 0 , q \\gt 0 )\\) \u3068\u304a\u304f.\r\n\\[\\begin{align}\r\n\\ell _ 1 & : \\ px +3qy = 3 \\\\\r\n& \\text{\u2234} \\quad \\dfrac{px}{3} +qy = 1 , \\\\\r\n\\ell _ 2 & : \\ \\dfrac{px}{\\cos^2 \\theta} -\\dfrac{qy}{\\sin^2 \\theta} = 2 \\\\\r\n& \\text{\u2234} \\quad \\dfrac{px}{2\\cos^2 \\theta} -\\dfrac{qy}{2\\sin^2 \\theta} = 1\n\\end{align}\\]\r\n\u306a\u306e\u3067\r\n\\[\\begin{align}\r\n\\text{Q} \\ \\left( 0 , \\dfrac{1}{q} \\right) , \\quad \\text{R} \\ \\left( 0 , -\\dfrac{2\\sin^2 \\theta}{q} \\right) \\quad ... [1]\n\\end{align}\\]\r\n\u307e\u305f, P \u306f \\(C _ 1 , C _ 2\\) \u306e\u4ea4\u70b9\u306a\u306e\u3067,\r\n\\[\r\np^2 + 3q^2 = 3 , \\quad \\dfrac{p^2}{\\cos^2 \\theta} - \\dfrac{q^2}{\\sin^2 \\theta} = 2\n\\]\r\n\\(p\\) \u3092\u6d88\u53bb\u3057\u3066\r\n\\[\r\n\\dfrac{3 \\left( 1-q^2 \\right)}{\\cos^2 \\theta} -\\dfrac{q^2}{\\sin^2 \\theta} = 2\n\\]\r\n\\(0 \\lt \\theta \\lt \\dfrac{\\pi}{2}\\) \u306b\u304a\u3044\u3066, \\(0 \\lt \\sin^2 \\theta \\lt 1 , \\ 0 \\lt \\cos^2 \\theta \\lt 1\\) \u306a\u306e\u3067, \u8fba\u3005\u306b \\(\\sin^2 \\theta \\cos^2 \\theta\\) \u3092\u639b\u3051\u3066\r\n\\[\\begin{align}\r\n3 \\left( 1-q^2 \\right) \\sin^2 \\theta -q^2 \\cos^2 \\theta & = 2 \\sin^2 \\theta \\cos^2 \\theta \\\\\r\n\\left( 3 \\sin^2 \\theta +\\cos^2 \\theta \\right) q^2 & = 3 \\sin^2 \\theta -2 \\sin^2 \\theta \\cos^2 \\theta \\\\\r\n\\left( 2 \\sin^2 \\theta +1 \\right) q^2 & = 3 \\sin^2 \\theta -2 \\sin^2 \\theta \\left( 1 -\\sin^2 \\theta \\right) \\\\\r\n\\left( 2 \\sin^2 \\theta +1 \\right) q^2 & = \\sin^2 \\theta \\left( 2 \\sin^2 \\theta +1 \\right)\n\\end{align}\\]\r\n\\(2 \\sin^2 \\theta +1 \\gt 0\\) \u306a\u306e\u3067, \u3053\u308c\u3067\u8fba\u3005\u3092\u5272\u308b\u3068\r\n\\[\r\nq^2 = \\sin^2 \\theta\n\\]\r\n\\(q \\gt 0\\) \u306a\u306e\u3067\r\n\\[\r\nq = \\sin \\theta\n\\]\r\n[1] \u306b\u7528\u3044\u308b\u3068\r\n\\[\r\n\\text{Q} \\ \\left( 0 , \\dfrac{1}{\\sin \\theta} \\right) , \\quad \\text{R} \\ \\left( 0 , -2\\sin \\theta \\right)\n\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\r\n\\text{QR} = \\dfrac{1}{\\sin \\theta} -\\left( -2\\sin \\theta \\right) = 2\\sin \\theta +\\dfrac{1}{\\sin \\theta}\n\\]\r\n\u76f8\u52a0\u76f8\u4e57\u5e73\u5747\u306e\u95a2\u4fc2\u3088\u308a,\r\n\\[\r\n\\text{QR} \\geqq 2 \\sqrt{2\\sin \\theta \\cdot \\dfrac{1}{\\sin \\theta}} = 2 \\sqrt{2}\n\\]\r\n\u7b49\u53f7\u6210\u7acb\u306f\r\n\\[\\begin{align}\r\n\\dfrac{1}{\\sin \\theta} & = 2\\sin \\theta \\\\\r\n\\sin \\theta & = \\dfrac{1}{\\sqrt{2}} \\\\\r\n\\text{\u2234} \\quad \\theta & = \\dfrac{\\pi}{4}\n\\end{align}\\]\r\n\u306e\u3068\u304d.
\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u6700\u5c0f\u5024\u306f\r\n\\[\r\n\\underline{2 \\sqrt{2}}\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(0 \\lt \\theta \\lt \\dfrac{\\pi}{2}\\) \u3068\u3059\u308b. \\(2\\) \u3064\u306e\u66f2\u7dda \\[ C _ 1 : \\ x^2 + 3y^2 = 3 , \\quad C _ 2 : \\ \\dfrac{x^2}{ … \u7d9a\u304d\u3092\u8aad\u3080