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