Question

A light post standing at point P on a circular path of radius ‘x’ subtends an angle $\theta$ at some point Q on the path and PQ subtends an angle of 45º at any other point on the path.
What is height of the light post ?

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

Answer 1

The correct option is A $\sqrt{2}rtan\theta$


Let PM be the light post at point P of height h.
Now, PQ subtends an angle of 45º at any point (say A) on the circle. So, it will subtend an angle of 90º at the center.
∵ ΔOPQ is right angled, triangle with right angle at O.
Applying pythagoras theorem.
$O{P}^2+O{Q}^2=P{Q}^2$
Now, OP and OQ are radius,
So, OP = OQ = r
We get, $r^2+r^2=P{Q}^2$
$\Rightarrow 2r^2=P{Q}^2$
$\Rightarrow PQ=\sqrt{2}r$
Now in $\Delta PMQ$
$tan\theta =\frac{PM}{PQ}$
$\Rightarrow h=\sqrt{2}rtan\theta$

So, correct answer is option a.