\u53cc\u66f2\u7dda $y = \\dfrac{1}{x}$ \u306e\u7b2c $1$ \u8c61\u9650\u306b\u3042\u308b\u90e8\u5206\u3068, \u539f\u70b9 O \u3092\u4e2d\u5fc3\u3068\u3059\u308b\u5186\u306e\u7b2c $1$ \u8c61\u9650\u306b\u3042\u308b\u90e8\u5206\u3092,\r\n\u305d\u308c\u305e\u308c $C _ 1 , C _ 2$ \u3068\u3059\u308b. $C _ 1$ \u3068 $C _ 2$ \u306f $2$ \u3064\u306e\u7570\u306a\u308b\u70b9 A, B \u3067\u4ea4\u308f\u308a, \u70b9 A \u306b\u304a\u3051\u308b $C _ 1$ \u306e\u63a5\u7dda $l$ \u3068\u7dda\u5206 OA \u306e\u306a\u3059\u89d2\u306f $\\dfrac{\\pi}{6}$ \u3067\u3042\u308b\u3068\u3059\u308b.\r\n\u3053\u306e\u3068\u304d, $C _ 1$ \u3068 $C _ 2$ \u3067\u56f2\u307e\u308c\u308b\u56f3\u5f62\u306e\u9762\u7a4d\u3092\u6c42\u3081\u3088. <\/p>\r\n
$C _ 1 , C _ 2$ \u306f, \u76f4\u7dda $y=x$ \u306b\u3064\u3044\u3066\u5bfe\u79f0\u306a\u306e\u3067, A $\\left( a , \\dfrac{1}{a} \\right)$ , B $\\left( \\dfrac{1}{a} , a \\right) \\ ( 0 \\lt a \\lt 1 )$ \u3068\u304a\u3044\u3066\u3082\u3088\u3044.
\r\n$l$ , OA \u305d\u308c\u305e\u308c\u304c $x$ \u8ef8\u6b63\u65b9\u5411\u3068\u306a\u3059\u89d2\u3092 $\\theta _ 1 , \\theta _ 2$ \u3068\u304a\u304f\u3068, \u6761\u4ef6\u3088\u308a\r\n$$\r\n\\theta _ 1 -\\theta _ 2 = \\dfrac{\\pi}{6}\r\n\\quad ... \\maru{1}\r\n$$\r\n$C$ \u306e\u5f0f\u3088\u308a, $y' = -\\dfrac{1}{x^2}$ \u306a\u306e\u3067\r\n$$\r\n\\tan \\theta _ 1 = -\\dfrac{1}{a^2}\r\n$$\r\n\u307e\u305f, \u6761\u4ef6\u3088\u308a\r\n$$\r\n\\tan \\theta _ 2 = \\dfrac{\\frac{1}{a}}{a} = \\dfrac{1}{a^2} \\quad ... \\maru{2}\r\n$$\r\n\u3057\u305f\u304c\u3063\u3066, $l$ \u3068 OA \u306f, \u76f4\u7dda $x=a$ \u306b\u3064\u3044\u3066\u5bfe\u79f0\u306a\u306e\u3067, $\\maru{1}$ \u3092\u7528\u3044\u3066\r\n$$\r\n\\theta _ 2 = \\dfrac{\\pi}{2} -\\dfrac{1}{2}\\cdot \\dfrac{\\pi}{6} = \\dfrac{5 \\pi}{12}\r\n$$\r\n$\\dfrac{5 \\pi}{12} = \\dfrac{\\pi}{4} +\\dfrac{\\pi}{6}$ \u306a\u306e\u3067, $\\tan$ \u306e\u52a0\u6cd5\u5b9a\u7406\u3088\u308a\r\n$$\\begin{align}\r\n\\tan \\dfrac{5 \\pi}{12} & = \\dfrac{1 +\\frac{1}{\\sqrt{3}}}{1 -1 \\cdot \\frac{1}{\\sqrt{3}}} = \\dfrac{\\sqrt{3} +1}{\\sqrt{3} -1} \\\\\r\n& = \\dfrac{2}{\\left( \\sqrt{3} -1 \\right)^2} = \\dfrac{\\left( \\sqrt{3} +1 \\right)^2}{2} = \\dfrac{1}{2 -\\sqrt{3}}\r\n\\end{align}$$\r\n\u3086\u3048\u306b, $\\maru{2}$ \u3088\u308a\r\n$$\r\na = \\dfrac{\\sqrt{3} -1}{\\sqrt{2}} , \\ \\dfrac{1}{a} = \\dfrac{\\sqrt{3} +1}{\\sqrt{2}}\r\n$$\r\n\u70b9 A, B \u304b\u3089 $x$ \u8ef8\u306b\u4e0b\u308d\u3057\u305f\u5782\u7dda\u306e\u8db3\u3092\u70b9 A', B' \u3068\u304a\u304d, $C _ 1$ \u3068\u7dda\u5206 AB \u306b\u56f2\u307e\u308c\u305f\u90e8\u5206\u306e\u9762\u7a4d\u3092 $T$ \u3068\u3059\u308b.
\r\n$\\triangle \\text{OAA'} = \\triangle \\text{OBB'}$ , $\\angle \\text{AOB} = \\dfrac{5 \\pi}{12} -\\dfrac{\\pi}{12} = \\dfrac{\\pi}{3}$ \u3067\u3042\u308b\u3053\u3068\u3092\u7528\u3044\u308c\u3070, \u6c42\u3081\u308b\u9762\u7a4d $S$ \u306f\r\n$$\\begin{align}\r\nS & = ( \\text{\u6247\u5f62 OAB} ) -\\triangle \\text{OAB} +T \\\\\r\n& = \\dfrac{1}{2} \\cdot \\text{OA}^2 \\cdot \\dfrac{\\pi}{3} -\\displaystyle\\int _ a^{\\frac{1}{a}} \\dfrac{1}{x} \\, dx \\\\\r\n& = \\dfrac{\\pi}{6} \\left( a^2 +\\dfrac{1}{a^2} \\right) -\\left[ \\log x \\right] _ a^{\\frac{1}{a}} \\\\\r\n& = \\dfrac{\\pi}{6} \\left\\{ \\dfrac{\\left( \\sqrt{3} -1 \\right)^2}{2} +\\dfrac{\\left( \\sqrt{3} +1 \\right)^2}{2} \\right\\} +2 \\log a \\\\\r\n& = \\dfrac{\\pi}{6} \\cdot 4 +\\log a^2 \\\\\r\n& = \\underline{\\dfrac{2 \\pi}{3} +\\log \\left( 2 -\\sqrt{3} \\right)}\r\n\\end{align}$$<\/p>","protected":false},"excerpt":{"rendered":"\u53cc\u66f2\u7dda $y = \\dfrac{1}{x}$ \u306e\u7b2c $1$ \u8c61\u9650\u306b\u3042\u308b\u90e8\u5206\u3068, \u539f\u70b9 O \u3092\u4e2d\u5fc3\u3068\u3059\u308b\u5186\u306e\u7b2c $1$ \u8c61\u9650\u306b\u3042\u308b\u90e8\u5206\u3092, \u305d\u308c\u305e\u308c $C _ 1 , C _ 2$ \u3068\u3059\u308b. $C _ 1$ \u3068 $C _ … \u7d9a\u304d\u3092\u8aad\u3080