Explanation

(b)
Given , $\frac{x^{2}}{2} - \frac{y^{2}}{1} = 1$
Here, $a^{2} = 2,\ b^{2} = 1$
Equation of asymptotes to the given hyperbola is
$\frac{x}{\sqrt{2}} - \frac{y}{1} = 0$ and $\frac{x}{\sqrt{2}} + \frac{y}{1} = 0$
Let $P(\sqrt{2}\sec{\theta,\tan{\theta)}\ }$ be any point, then product of length of perpendicular.
$= \frac{\left\lbrack \frac{\sqrt{2\ \ }\sec\theta}{\sqrt{2}} - \frac{\tan\theta}{1} \right\rbrack\ \left\lbrack \frac{\sqrt{2}\sec\theta}{\sqrt{2}} + \frac{\tan\theta}{1} \right\rbrack}{\sqrt{\frac{1}{2} + \frac{1}{1}}\ \sqrt{\frac{1}{2} + \frac{1}{1}}}\$
$= \frac{\sec^{2}\theta - \tan^{2}\theta}{\frac{3}{2}\ }$
$= \frac{2}{3}$