At a point on a level plane a tower subtends an angle θ

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At a point on a level plane a tower subtends an angle θ and a flag-staff a ft. in length at the top of the tower subtends an angle ɸ. The height of the tower is :

1. asin θ cos ɸ / cos ( θ + ɸ )
2. asin θ cos ( θ + ɸ ) / sin ɸ
3. a cos ( θ + ɸ ) / sin θ sin ɸ
4. None of these

Correct Option: D

Let OP be the tower of height h (say) and PQ the flag-staff of height a, such that
∠OAP = θ and ∠PAQ = ɸ
In ΔOAP and ∠OAQ
OA = OP cot θ = h cotθ
and OA = OQ cot ( θ + ɸ ) = (h + a) cot ( θ + ɸ )
∴ h cotθ = (h + a) cot ( θ + ɸ )
⇒ h = a cot ( θ + ɸ ) / cotθ - cot ( θ + ɸ )