A light post standing at point P on a circular path of radius ‘x’ subtends an angle θ 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
√2rtanθ
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B
√2rtanθ
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C
√2rcotθ
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D
r√2cotθ
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Solution
The correct option is A√2rtanθ
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. OP2+OQ2=PQ2
Now, OP and OQ are radius,
So, OP = OQ = r
We get, r2+r2=PQ2 ⇒2r2=PQ2 ⇒PQ=√2r
Now in ΔPMQ tanθ=PMPQ ⇒h=√2rtanθ