Question

A man standing on a horizontal plane, observes the angle of elevation of the top of a tower to be $\alpha$. After waking a distance equal to double the height of the tower, the angle of elevation becomes $2\alpha$ then $\alpha$ is

• A. $\frac{\pi }{12}$
• B. $\frac{\pi }{18}$
• C. $\frac{\pi }{6}$
• D. $\pi$

Answer 1

The correct option is A $\frac{\pi }{12}$

$\tan \alpha =\frac{h}{2h+x},\tan 2\alpha =\frac{h}{x}\Rightarrow x=\frac{h}{\tan 2\alpha }$

$\tan \alpha =\frac{h}{2h+\frac{h}{\tan 2\alpha }}=\frac{1}{2+\frac{1}{\tan 2\alpha }}$

$\tan \alpha =\frac{1}{\frac{2\tan 2\alpha +1}{\tan 2\alpha }}=\frac{\tan 2\alpha }{1+2\tan 2\alpha }$

$\tan \alpha =\frac{\frac{2\tan \alpha }{1-{\tan }^2\alpha }}{1+\frac{4\tan \alpha }{1-{\tan }^2\alpha }}$

$\tan \alpha =\frac{2\tan \alpha }{1-{\tan }^2\alpha +4\tan \alpha }$

$1-{\tan }^2\alpha +4\tan \alpha =2$

${\tan }^2\alpha -4\tan \alpha +1=0$

$\tan \alpha =\frac{4\pm \sqrt{16-4}}{2}$

$=\frac{4\pm 2\sqrt{3}}{2}$

$\tan \alpha =2\pm \sqrt{3}$

$\alpha ={15}^o=\frac{\pi }{12}$

Hence, the option $A$ is the correct answer.