Question

The angle of elevation of the top of a tower as seen from two points $A$ & $B$ situated the same line and at distance '$p$' and '$q$' respectively from the foot of the tower are complementary, then height of the tower is

• A. $pq$
• B. $\frac{p}{q}$
• C. $\sqrt{pq}$
• D. none of these

Answer 1

The correct option is C $\sqrt{pq}$

Let $OT$ be the tower of height $h$

Let $O$ be the base of the tower

Let $A$ and $B$ be two points on the through the base such that $OA=p$ and $OB=q$

In $\Delta AOT$, we have

$\tan \alpha =\frac{OT}{OA}$

$=\frac{h}{p}$ .....(i)

In $\Delta BOT$, we have

$\tan (90-\alpha )=\frac{OT}{OB}$

$=\frac{h}{q}$

$\cot \alpha =\frac{h}{q}$ ......(ii)

Multiplying (i) and (ii), we have

$\Rightarrow \tan \alpha \cot \alpha =hp\times hq$

$\Rightarrow \tan \alpha \frac{1}{\tan \alpha }=\frac{h^2}{pq}$ ......... [since $cot=\frac{1}{tan}$]

$\Rightarrow 1=\frac{h^2}{pq}$

$\Rightarrow h^2=pq$

$\Rightarrow h=\sqrt{pq}$