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For any real t, $x = \frac{e^{t} + e^{- t}}{2}, y = \frac{e^{t} - e^{- t}}{2}$ is a point on the hyperbola $x^{2} - y^{2} = 1$ then area bounded by hyperbola and lines joining its centre to points corresponding to t and –t is

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Detailed Solution

Area of $ΔOAB = xy$
Area b/w line and arc AB of curve $= 2 \int_{1}^{x} \sqrt{x^{2} - 1} = xy - \log (x + y)$
$∴$ Area of shaded region $= \log (x + y) = \log e^{t} = t$

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