Question

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 metre is __________.

• A. $\frac{a}{4}$
• B. $\frac{a}{\sqrt{2}}$
• C. $a\sqrt{2}$
• D. $\frac{a}{2\sqrt{2}}$

Answer 1

Answer: (D)

$\frac{a}{2\sqrt{2}}$

Let AB $=a$ be the line joining feet of the persons and C be the mid-point of this line.

Also, let $h$ be the height of the shorter person, then height of the taller person will be $2h$.
Let angle of elevations be $x$ and $90-x$ as they are given complementary.

From figure,

$\tan x=\frac{h}{\frac{a}{2}}=\frac{2h}{a}....\left(1\right)$

Also,

$\tan (90-x)=\frac{2h}{\frac{a}{2}}=\frac{4h}{a}=\cot x....\left(2\right)$

Multiplying $\left(1\right)$ and $2$, we get

$\tan x\times \cot x=\frac{8h^2}{a^2}=1$

$\therefore a^2=8h^2$

$\Rightarrow h=\frac{a}{2\sqrt{2}}$