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From a point at a height $h$ m above a lake, the angle of elevation of a cloud is $α$ and the angle of depression of its reflection in the lake is $β$ . What will be the height of the cloud above the surface of the lake?

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\frac{hcos (α + β)}{\cos (α - β)} m

\frac{hsin (α + β)}{\cos (α - β)} m

\frac{hsin (α - β)}{\cos (α + β)} m

\frac{hsin (α + β)}{\sin (β - α)} m

answer is D.

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Detailed Solution

It is given that from a point at a height

h

m above a lake, the angle of elevation of a cloud is

α

and the angle of depression of its reflection in the lake is

β

Let

H

be the height of the cloud from the point above

h m

from the lake surface.
Now the distance of the reflection from the lake surface be

H + h m

.
Now, In

△ CDE,

tanα = \frac{DE}{CE}

⇒

CE = \frac{DE}{tanα} = \frac{H}{tanα} \dots . (1)

△ CE D^{'},

tanβ = \frac{E D^{'}}{CE}

⇒

CE = \frac{E D^{'}}{tanβ} = \frac{H + 2 h}{tanβ} \dots . (2)

From equation

(1) (2),

we get,

\frac{H}{tanα} = \frac{H + 2 h}{tanβ}

⇒

Htanβ = Htanα + 2 htanα

⇒

H (tanβ - tanα) = 2 htanα

⇒

H = \frac{2 htanα}{tanβ - tanα}

Now, height of the cloud above the lake surface,
⇒

H + h = \frac{2 htanα}{tanβ - tanα} + h

⇒

H = \frac{2 htanα + htanβ - htanα}{tanβ - tanα}

⇒

H = \frac{h (tanα + tanβ)}{tanβ - tanα}

⇒

H = h [\frac{\frac{sinα}{cosα} + \frac{sinβ}{cosβ}}{\frac{sinβ}{cosβ} - \frac{sinα}{cosα}}]

⇒

H = h [\frac{\frac{sinαcosβ + cosαsinβ}{cosαcosβ}}{\frac{sinβcosα - cosβsinα}{cosαcosβ}}]

⇒

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

Hence, the correct option is 4.

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