Question

The angle of elevation of the top of a tower from two points distant $a$ and $b$ from the base and in the same straight line with it are complementary. The height of the tower is

• A. $\sqrt{\frac{a}{b}}$.
• B. $\sqrt{ab}$.
• C. $ab$.
• D. None of these

Answer 1

The correct option is B $\sqrt{ab}$.

Let $AB$ be the height of the tower and $B$ be its foot of the perpendicular

$AB=h,BE=b,BC=a$

$\angle AEC={60}^{\circ },\angle ACB={30}^{\circ }$

In $△ABC,\tan {30}^{\circ }=\frac{AB}{BC}$

$\Rightarrow \frac{1}{\sqrt{3}}=\frac{h}{a}$

$\Rightarrow h=\frac{a}{\sqrt{3}}$ .......$\left(1\right)$

In $△AEB,\tan {60}^{\circ }=\frac{AB}{BE}$

$\Rightarrow \sqrt{3}=\frac{h}{b}$

$\Rightarrow h=b\sqrt{3}$ .......$\left(2\right)$

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

$h^2=\frac{a}{\sqrt{3}}\times b\sqrt{3}=ab$

$\therefore h=\sqrt{ab}$