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The angle of elevation of a cloud from a point $h$ $m$ above the level of water in a lake is $α$ and the angle of depression of its reflection in the lake is $β$ . Then height of the cloud above the water level is

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\frac{hsin (β - α)}{\sin (β + α)}

\frac{hsin (β - α)}{\sin (β - α)}

\frac{hsin (α + β)}{\sin (β - α)}

none of these

answer is C.

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Detailed Solution

Given that,
The height of the point above the water level = h m
Angle of elevation of a cloud from the point =

α

Angle of depression of its reflection in the lake =

β

We have the formula,

tanθ = \frac{Height}{Distance}

We have to find the height of the cloud.
In

ΔACP,

tan α = \frac{PC}{AC}

tan α = \frac{H - h}{x}

⟹ x = \frac{H - h}{tanα}

ΔAC P^{'}

tanβ = \frac{C P^{'}}{AC}

tanβ = \frac{H + h}{x}

⟹ x = \frac{H + h}{tanβ}

⟹ \frac{H - h}{tanα} = \frac{H + h}{tanβ}

⟹ Htanβ - htanβ = Htanα + htanα

⟹ H (tanβ - tanα) = h (tanα + tanβ)

⟹ H = h \frac{tanα + tanβ}{tanβ - tanα}

⟹ H = h \frac{sinαcosβ + sinβcosα}{sinβcosα - sinαcosβ}

⟹ H = \frac{hsin (α + β)}{\sin (β - α)}

Thus, the height of the cloud is

\frac{hsin (α + β)}{\sin (β - α)}

m.
Therefore, the correct option is 3.

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