Explanation

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