Question

A lamp post standing at a point $A$ on a circular path of radius $r$ subtends an angle $\alpha$ at some point $B$ on the path, and $AB$ subtends an angle of ${45}^{\circ }$ at any other point on the path, the height of the lamp post is

• A. $\sqrt{2}r\cot \alpha$
• B. $\frac{r}{\sqrt{2}}\tan \alpha$
• C. $\sqrt{2}r\tan \alpha$
• D. $\frac{r}{\sqrt{2}}\cot \alpha$

Answer 1

The correct option is C $\sqrt{2}r\tan \alpha$

Let $AM$ is the lamp post at point $A$ of height $h.$

$AB$ subtends an angle of ${45}^{\circ }$ at any point on the circle. Therefore it will subtend an angle of ${90}^{\circ }$ at the center.

So, we can say that $△OAB$ is a right-angled triangle with a right angle at $O.$

By applying pythagoras theorem,

$\begin{array}{cc}O{A}^2+O{B}^2& =A{B}^2\\ r^2+r^2& =A{B}^2\\ A{B}^2& =2r^2\\ AB& =r\sqrt{2}\end{array}$

In $△AMB,$
$\tan \alpha =\frac{AM}{AB}$

$\Rightarrow \tan \alpha =\frac{h}{r\sqrt{2}}$

$\Rightarrow h=r\sqrt{2}\tan \alpha$