Question

If $ \tan^{-1}(a/x)+\tan^{-1}(b/x) $ = $ \pi /2 $ , then $ x $ is equal to

• A. $ \sqrt{\frac{b}{a}} $
• B. $ \sqrt{\frac{a}{b}} $
• C. $ -\sqrt{ab} $
• D. $ \sqrt{ab} $

Answer 1

Answer: (D)

Given, $\tan ^{-1}\left(\frac{a}{x}\right)+\tan ^{-1}\left(\frac{b}{x}\right)=\frac{\pi}{2}$
$\Rightarrow \tan ^{-1}\left(\frac{\frac{a}{x}+\frac{b}{x}}{1-\frac{a}{x} \times \frac{b}{x}}\right)=\frac{\pi}{2}$
$\Rightarrow \frac{(a+b) x}{x^{2}-a b}=\tan \frac{\pi}{2}=\frac{1}{0}$
$ \Rightarrow x^{2} =a b$
$ \Rightarrow x =\pm \sqrt{a b} $
$ \Rightarrow x =\sqrt{a b} $
$(x=-\sqrt{a b} $ does not satisfy the given equation)