Question

The angle of elevation of the top of pole from the point $D$ on the ground be $\alpha$, whereas the angle of elevation of the top of pole from the point which is at a height ${}^{\prime }{b}^{\prime }$ above point $D$ is $\beta$, then the length of pole is:

• A. $\frac{b\cot \beta \tan \alpha }{\cot \beta \tan \alpha -1}$
• B. $b\cot \alpha .\tan \beta$
• C. $b\tan \alpha .\tan \beta$
• D. $b\cot \alpha .\cot \beta$

Answer 1

Answer: (A)

$\frac{b\cot \beta \tan \alpha }{\cot \beta \tan \alpha -1}$

Let $AB$ be the Pole. C be the point at a height 'b' above point D.

Also, $DC=b=AE,AD=CE$, {from geometry}

Let say $BE=x$

In $△BCE,$

$\frac{CE}{x}=\cot \beta$

$CE=x\cot \beta =AD$ .........(1)

In $△ADB,$

$\tan \alpha =\frac{AB}{AD}=\frac{AE+BE}{AD}=\frac{x+b}{x\cot \beta }$ {From (1)}

$\left(\tan \alpha \right)x\cot \beta =x+b$

$x=\frac{b}{\cot \beta \tan \alpha -1}$

$AB=x+b=\frac{b\cot \beta \tan \alpha }{\cot \beta \tan \alpha -1}$