Explanation

(a)
Given, $\tan^{- 1}{\left( \frac{a}{x} \right) + \tan^{- 1}{\left( \frac{b}{x} \right) = \frac{\pi}{2}}}$
$\Rightarrow \ \tan^{- 1}\left( \frac{\frac{a}{x} + \frac{b}{x}}{1 - \frac{ab}{x^{2}}} \right) = \frac{\pi}{2}$
$\Rightarrow \ \ \ \ \ \ \ \frac{(a + b)x}{x^{2} - ab}$
$\ \ \ \ \ \ \ \ \ \ \ \ \ = \tan\frac{\pi}{2}$
$\Rightarrow \ \ \frac{(a + b)x}{x^{2} - ab} = \frac{1}{0}$
$\Rightarrow \ \ x^{2} - ab = 0$
$\Rightarrow \ \ \ x = \sqrt{ab}$