If the angle of elevation of a cloud from a point h metres above a lake is alpha and the angle of depression of its reflection is beta. Prove that the height of cloud is h(tan beta + tan alpha)/ tan beta - tan alpha

Let AN be the surface of the lake and O be the point of observation such that OA = h metres.

Let P be the position of the cloud and P' be its reflection in the lake 

Then PN = P'N

Let OM ⊥ PN

Also, ∠POM = α and ∠P'OM = β

Let PM = x

Then PN = PM + MN = PM + OA = x + h

In rt. ΔOPM, we have

In rt.  ΔOMP', we have,

Equating (1) and (2):

Hence, height of the cloud is given by PN = x + h

Hence proved.

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