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From the top of a cliff of height $a$ , the angle of depression of the foot of a certain tower is found to be double the angle of elevation of the top of the tower of height $h$ . If $θ$ be the angle of elevation then its value is?

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\cos^{- 1} \sqrt{\frac{2 h}{a}}

\sin^{- 1} \sqrt{\frac{2 h}{a}}

\sin^{- 1} \sqrt{\frac{a}{2 - h}}

\tan^{- 1} \sqrt{3 - \frac{2 h}{a}}

answer is D.

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Detailed Solution

It is given that from the top of a cliff of height

a

, the angle of depression of the foot of a certain tower is found to be double the angle of elevation of the top of the tower of height

h

θ

be the angle of elevation.
Let AB be the tower and PQ be the cliff.
Question Image

△ QBM

h - a = xtanθ . . . . . (1)

△ QAP

a = xtan 2 θ . . . . . . . . (2)

Dividing 1 by 2, we get,
⇒

\frac{h - a}{a} = \frac{tanθ}{\tan 2 θ}

⇒

\tan 2 θ = \frac{2 tanθ}{1 - \tan^{2} θ}

⇒

\frac{h}{a} - 1 = \frac{1 - \tan^{2} θ}{2}

⇒

1 - \tan^{2} θ = \frac{2 h}{a} - 2

⇒

\tan^{2} θ = 3 - \frac{2 h}{a}

⇒

tanθ = \sqrt{3 - \frac{2 h}{a}}

⇒

θ = \tan^{- 1} (\sqrt{3 - \frac{2 h}{a}})

Hence, the correct option is 4.

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