\u4e0d\u7b49\u5f0f \\(1 \\leqq x^2+y^2 \\leqq 4\\) \u304c\u8868\u3059 \\(xy\\) \u5e73\u9762\u5185\u306e\u9818\u57df\u3092 \\(D\\) \u3068\u3059\u308b.\r\nP \u3092\u5186 \\(x^2+y^2 = 1\\) \u4e0a\u306e\u70b9, Q \u3068 R \u3092\u5186 \\(x^2+y^2 = 4\\) \u4e0a\u306e\u7570\u306a\u308b \\(2\\) \u70b9\u3068\u3057, \u4e09\u89d2\u5f62 PQR \u306f\u9818\u57df \\(D\\) \u306b\u542b\u307e\u308c\u3066\u3044\u308b\u3068\u3059\u308b.\r\n\\(a , b\\) \u3092\u5b9f\u6570\u3068\u3057, \u884c\u5217 \\(A = \\left( \\begin{array}{cc} a & -b \\\\ b & a \\end{array} \\right)\\) \u306e\u8868\u3059 \\(1\\) \u6b21\u5909\u63db\u306b\u3088\u308a P \u306f P' , Q \u306f Q' , R \u306f R' \u306b\u79fb\u3055\u308c\u308b.\r\n\u3053\u306e\u3068\u304d ,\u4e09\u89d2\u5f62 P'Q'R' \u304c\u9818\u57df \\(D\\) \u306b\u542b\u307e\u308c\u308b\u305f\u3081\u306e \\(a , b\\) \u306e\u5fc5\u8981\u5341\u5206\u6761\u4ef6\u3092\u6c42\u3081\u3088. \u305f\u3060\u3057, \u4e09\u89d2\u5f62\u306f\u5185\u90e8\u3082\u542b\u3081\u3066\u8003\u3048\u308b\u3082\u306e\u3068\u3059\u308b.<\/p>\r\n
P \\(( 1 , 0 )\\) , Q \\(\\left( \\dfrac{3}{2} , \\dfrac{\\sqrt{7}}{2} \\right)\\) , R \\(\\left( \\dfrac{3}{2} , -\\dfrac{\\sqrt{7}}{2} \\right)\\) \u306e\u5834\u5408\u3092\u8003\u3048\u308b.
\r\n\u3053\u306e\u3068\u304d, \u25b3PQR \u306f \u9818\u57df \\(D\\) \u306b\u542b\u307e\u308c\u3066\u3044\u308b.
\r\nP' \u306e\u5ea7\u6a19\u306f\r\n\\[\r\n\\left( \\begin{array}{cc} a & -b \\\\ b & a \\end{array} \\right) \\left( \\begin{array}{c} 1 \\\\ 0 \\end{array} \\right) = \\left( \\begin{array}{c} a \\\\ b \\end{array} \\right) \\ .\r\n\\]\r\n\u3053\u308c\u304c, \u9818\u57df \\(D\\) \u306b\u542b\u307e\u308c\u308b\u306e\u3067\r\n\\[\r\n1 \\leqq a^2 +b^2 \\leqq 4 \\quad ... [1] \\ .\r\n\\]\r\n\u307e\u305f, Q' , R' \u306e\u5ea7\u6a19\u306f\r\n\\[\r\n\\left( \\begin{array}{cc} a & -b \\\\ b & a \\end{array} \\right) \\left( \\begin{array}{c} \\dfrac{3}{2} \\\\ \\pm \\dfrac{\\sqrt{7}}{2} \\end{array} \\right) = \\left( \\begin{array}{c} \\dfrac{3a \\mp \\sqrt{7} b}{2} \\\\ \\dfrac{3b \\pm \\sqrt{7} a}{2} \\end{array} \\right) \\ .\r\n\\]\r\n\u3053\u308c\u3082, \u9818\u57df \\(D\\) \u306b\u542b\u307e\u308c\u308b\u306e\u3067\r\n\\[\\begin{align}\r\n1 \\leqq \\left( \\dfrac{3a \\mp \\sqrt{7} b}{2} \\right)^2 & +\\left( \\dfrac{3b \\pm \\sqrt{7} a}{2} \\right)^2 \\leqq 4 \\\\\r\n1 \\leqq 4 a^2 & +4 b^2 \\leqq 4 \\\\\r\n\\text{\u2234} \\quad \\dfrac{1}{4} \\leqq a^2 & +b^2 \\leqq 1 \\quad ... [2] \\ .\r\n\\end{align}\\]\r\n[1] [2] \u3088\u308a\r\n\\[\r\na^2 +b^2 = 1 \\quad ... [3] \\ .\r\n\\]\r\n\u3067\u3042\u308b\u3053\u3068\u304c\u5fc5\u8981\u3067\u3042\u308b.
\r\n\u3053\u306e\u3068\u304d, \\(a = \\cos \\theta\\) , \\(b = \\sin \\theta\\) \u3068\u304a\u304f\u3053\u3068\u304c\u3067\u304d\u3066\r\n\\[\r\nA = \\left( \\begin{array}{cc} \\cos \\theta & -\\sin \\theta \\\\ \\sin \\theta & \\cos \\theta \\end{array} \\right) \\ .\r\n\\]\r\n\u306a\u306e\u3067, \\(A\\) \u306f\u539f\u70b9\u3092\u4e2d\u5fc3\u306b\u3057\u305f\u89d2 \\(\\theta\\) \u306e\u56de\u8ee2\u3092\u8868\u3059.
\r\n\u9818\u57df \\(D\\) \u306e\u5883\u754c\u306f, \u539f\u70b9\u3092\u4e2d\u5fc3\u3068\u3059\u308b\u5186\u306a\u306e\u3067, \u25b3PQR \u304c\u9818\u57df \\(D\\) \u306b\u542b\u307e\u308c\u308c\u3070, \u25b3P'Q'R' \u3082\u9818\u57df \\(D\\) \u306b\u542b\u307e\u308c\u308b.
\r\n\u3057\u305f\u304c\u3063\u3066, [3] \u306f\u5341\u5206\u6761\u4ef6\u3067\u3082\u3042\u308b.
\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u6761\u4ef6\u306f\r\n\\[\r\n\\underline{a^2 +b^2 = 1} \\ .\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u4e0d\u7b49\u5f0f \\(1 \\leqq x^2+y^2 \\leqq 4\\) \u304c\u8868\u3059 \\(xy\\) \u5e73\u9762\u5185\u306e\u9818\u57df\u3092 \\(D\\) \u3068\u3059\u308b. P \u3092\u5186 \\(x^2+y^2 = 1\\) \u4e0a\u306e\u70b9, Q \u3068 R \u3092\u5186 \\(x^2+y^2 = 4 … \u7d9a\u304d\u3092\u8aad\u3080