$\tan^{-1}\big(\large\frac{a}{x}\big)$$+\tan^{-1}\big(\large\frac{b}{x}\big)=\large\frac{\pi}{2}$
$\tan^{-1}\big(\large\frac{a}{x}\big)$$+\cot^{-1}\big(\large\frac{x}{b}\big)=\frac{\pi}{2}$
$\large\frac{a}{x}=\frac{x}{b}$
$x^2=ab$
$x=\pm\sqrt{ab}$
Hence (d) is the correct option.