question_answer

Two persons are 'a' metres apart and the height of one is double that of the other. If from the middle point of the line joining their feet, an observer finds the angular elevation of their tops to be complementary, then the height of the shorter person in metres is ______.

A)  \[\frac{a}{4}\]                                   

B)  \[\frac{a}{\sqrt{2}}\]            

C)         \[a\sqrt{2}\]   

D)         \[\frac{a}{2\sqrt{2}}\]

Correct Answer: D

Solution :

Let the height of shorter person be x, and the height of 2nd one be 2x. \[{{\theta }_{1}}+{{\theta }_{2}}={{90}^{o}}\] (Given)                \[\therefore \]    \[\tan {{\theta }_{1}}\times \tan {{\theta }_{2}}=1\] Now,   \[\tan {{\theta }_{1}}=\frac{2x}{a/2}\] and  \[\tan {{\theta }_{2}}=\frac{x}{a/2}\] \[\therefore \] \[\frac{2x}{a/2}\times \frac{x}{a/2}=1\Rightarrow {{x}^{2}}=\frac{{{a}^{2}}}{4\times 2}\Rightarrow x=\frac{a}{2\sqrt{2}}.\]