Question

A tower subtends an angle $\alpha$ at a point on the same level as the root of the tower and at a second point, $b$ meters above the first, the angle of depression of the foot of the tower is $\beta$. The height of the tower is

• A. $b\cot \alpha \tan \beta$
• B. $b\tan \alpha \tan \beta$
• C. $b\tan \alpha \cot \beta$
• D. None of these

Answer 1

Answer: (C)

$b\tan \alpha \cot \beta$

From the figure, in $△ABD$

$\frac{AB}{BD}=\tan \alpha$

$\Rightarrow \frac{h}{x}=\tan \alpha$

$\Rightarrow h=x\tan \alpha \dots \left(i\right)$

In right angled $△BCE$

$\frac{BE}{EC}=\tan \beta$

$\Rightarrow \frac{b}{x}=\tan \beta$

$\Rightarrow x=\frac{b}{\tan \beta }\dots \left(ii\right)$

Now, from equation $\left(i\right)$ and $\left(ii\right),$

$h=\frac{b}{\tan \beta }\times \tan \alpha$

$=b\tan \alpha \cot \beta$

Hence, height of the tower is equal to $b\tan \alpha \cot \beta .$