\u5b9a\u6570 \\(k\\) \u306f \\(k \\gt 1\\) \u3092\u307f\u305f\u3059\u3068\u3059\u308b. \\(xy\\) \u5e73\u9762\u4e0a\u306e\u70b9 A \\(( 1 , 0 )\\) \u3092\u901a\u308a \\(x\\) \u8ef8\u306b\u5782\u76f4\u306a\u76f4\u7dda\u306e\u7b2c \\(1\\) \u8c61\u9650\u306b\u542b\u307e\u308c\u308b\u90e8\u5206\u3092, \\(2\\) \u70b9 X, Y \u304c \\(\\text{AY} = k \\text{AX}\\) \u3092\u307f\u305f\u3057\u306a\u304c\u3089\u52d5\u3044\u3066\u3044\u308b.\r\n\u539f\u70b9 O \\(( 0 , 0 )\\) \u3092\u4e2d\u5fc3\u3068\u3059\u308b\u534a\u5f84 \\(1\\) \u306e\u5186\u3068\u7dda\u5206 OX, OY \u304c\u4ea4\u308f\u308b\u70b9\u3092\u305d\u308c\u305e\u308c P, Q \u3068\u3059\u308b\u3068\u304d, \u25b3OPQ \u306e\u9762\u7a4d\u306e\u6700\u5927\u5024\u3092 \\(k\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p>\r\n
X \\(( 1 , t ) \\ ( t \\gt 0 )\\) \u3068\u304a\u304f\u3068, Y \\(( 1 , kt )\\) .
\r\n\u307e\u305f\r\n\\[\r\n\\text{OX} : \\ y = tx , \\ \\text{OY} : \\ y = ktx\r\n\\]\r\n\u5186 \\(x^2+y^2 = 1\\) \u3068\u7dda\u5206 OX \u306e\u4ea4\u70b9 P \u306e\u5ea7\u6a19\u3092\u6c42\u3081\u308b\u3068\r\n\\[\\begin{align}\r\nx^2 & +(tx)^2 = 1 \\\\\r\n\\text{\u2234} \\quad x & = \\dfrac{1}{\\sqrt{t^2+1}} \\\\\r\n\\text{\u2234} \\quad y & = \\dfrac{t}{\\sqrt{t^2+1}}\r\n\\end{align}\\]\r\n\u306a\u306e\u3067\r\n\\[\r\n\\text{P} \\ \\left( \\dfrac{1}{\\sqrt{t^2+1}} , \\dfrac{t}{\\sqrt{t^2+1}} \\right)\r\n\\]\r\n\u540c\u69d8\u306b\u3059\u308c\u3070, \u5186 \\(x^2+y^2=1\\) \u3068\u7dda\u5206 OY \u306e\u4ea4\u70b9\u306f,\r\n\\[\r\n\\text{Q} \\ \\left( \\dfrac{1}{\\sqrt{k^2t^2+1}} , \\dfrac{kt}{\\sqrt{k^2t^2+1}} \\right)\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\\begin{align}\r\n\\triangle \\text{OPQ} & = \\dfrac{1}{2} \\left| \\dfrac{1}{\\sqrt{t^2+1}} \\cdot \\dfrac{kt}{\\sqrt{k^2t^2+1}} -\\dfrac{t}{\\sqrt{t^2+1}} \\cdot \\dfrac{1}{\\sqrt{k^2t^2+1}} \\right| \\\\\r\n& = \\dfrac{1}{2} \\cdot \\dfrac{(k-1)t}{\\sqrt{t^2+1}\\sqrt{k^2t^2+1}} \\quad ( \\ \\text{\u2235} \\ k \\gt 1 ) \\\\\r\n& = \\dfrac{k-1}{2} \\cdot \\dfrac{1}{\\sqrt{\\dfrac{1}{t^2}(t^2+1)(k^2t^2+1)}} \\\\\r\n& = \\dfrac{k-1}{2} \\cdot \\dfrac{1}{\\sqrt{k^2+1+k^2t^2+\\dfrac{1}{t^2}}} \\\\\r\n& \\leqq \\dfrac{k-1}{2} \\cdot \\dfrac{1}{\\sqrt{k^2+1+2\\sqrt{k^2t^2 \\cdot \\dfrac{1}{t^2}}}} \\quad ( \\ \\text{\u2235} \\ \\text{\u76f8\u52a0\u76f8\u4e57\u5e73\u5747\u306e\u95a2\u4fc2} ) \\\\\r\n& = \\dfrac{k-1}{2} \\cdot \\dfrac{1}{\\sqrt{k^2+2k+1}} = \\dfrac{k-1}{2(k+1)}\r\n\\end{align}\\]\r\n\u7b49\u53f7\u6210\u7acb\u306f, \\(k^2t^2 = \\dfrac{1}{t^2}\\) \u3059\u306a\u308f\u3061 \\(t=\\dfrac{1}{\\sqrt{k}}\\) \u306e\u3068\u304d\u3067\u3042\u308b.
\r\n\u3088\u3063\u3066, \u25b3OPQ \u306e\u9762\u7a4d\u306e\u6700\u5927\u5024\u306f\r\n\\[\r\n\\underline{\\dfrac{k-1}{2(k+1)}}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u5b9a\u6570 \\(k\\) \u306f \\(k \\gt 1\\) \u3092\u307f\u305f\u3059\u3068\u3059\u308b. \\(xy\\) \u5e73\u9762\u4e0a\u306e\u70b9 A \\(( 1 , 0 )\\) \u3092\u901a\u308a \\(x\\) \u8ef8\u306b\u5782\u76f4\u306a\u76f4\u7dda\u306e\u7b2c \\(1\\) \u8c61\u9650\u306b\u542b\u307e\u308c\u308b\u90e8\u5206\u3092, \\(2\\) \u70b9 X, … \u7d9a\u304d\u3092\u8aad\u3080