Question

The sum of angles 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 is $90^{\circ }$. Then, the height of the tower is

• A. $a^{2}b$
• B. $a{b}^2$
• C. $\sqrt{ab}$
• D. $ab$

Answer 1

Answer: (C)

Given, $\theta +\varphi =90^{\circ }$
Let the height of a tower $=h$
Then, In $\Delta ACP$
$\tan \theta =\frac{h}{a}$ ...(i)
In $\Delta ABP$
$\tan \varphi =\frac{h}{b}$ ...(ii)

$\because \tan (\theta +\varphi )=\frac{\tan \theta +\tan \varphi }{1-\tan \theta \cdot \tan \varphi }$
$\Rightarrow \tan 90^{\circ }=\frac{\frac{h}{a}+\frac{h}{b}}{1-\frac{h}{a}\cdot \frac{h}{b}}=\infty$
$\Rightarrow 1-\frac{h^2}{ab}=0$
$\Rightarrow h^2=ab$
$\Rightarrow h=\sqrt{ab}$
Hence, the required height of a tower is $\sqrt{ab}$.