Question

The angle of elevation of a cliff from a point $A$ on the ground and from the point $B$ $100m$ vertically above $A$ are $\alpha$ and $\beta$ respectively. The height of the cliff (in metres) is

• A. $\frac{100\tan \beta }{\cot \beta -\cot \alpha }$
• B. $\frac{100\cot \beta }{\cot \beta -\cot \alpha }$
• C. $\frac{100\tan \beta }{\cot \alpha +\cot \beta }$
• D. $\frac{100\cot \beta }{\cot \alpha +\cot \beta }$

Answer 1

The correct option is B $\frac{100\cot \beta }{\cot \beta -\cot \alpha }$

$\tan \alpha =\frac{h}{d}$

$\tan \beta =\frac{h-100}{d}$

$\Rightarrow \frac{\tan \alpha }{\tan \beta }=\frac{h}{h-100}$

$\Rightarrow \frac{\tan \alpha }{\tan \beta }=\frac{h}{h-100}$

$\Rightarrow \frac{\tan \beta }{\tan \alpha }=1-\frac{100}{h}$

$\Rightarrow \frac{\tan \alpha -\tan \beta }{\tan \alpha }=\frac{100}{h}$

$\Rightarrow h=\frac{100\tan \alpha }{\tan \alpha -\tan \beta }$
or

$h=\frac{100\cot \beta }{\cot \beta -\cot \alpha }$