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 in the take is `beta` , prove that the height of the cloud is `(h(tanbeta+t a nalpha)/(tanbeta-t a nalpha)`

Explanation

Let AB be the surface of the lake the let P be a point vertically above A such that `AP = h` metres.
Let C be the position of the cloud and let D be its reflection in the lake.
Draw `PQ_|_CD` . Then,
`angleQPC = alpha, angleQPD = beta`,
`BQ = AP = h` metres.
Let `CQ = x` metres. Then,
`BD = BC = (x+h)` metres.
From right `DeltaPQC` we have
`(PQ)/(CQ) = cot alpha rArr (PQ)/(x m ) = cot alpha`
`rArr PQ= x cot alpha ` metres. `"..........."(i)`
From right `DeltaPQD` , we have
`(PQ)/(QD) = cot beta rArr (PQ)/((x+2h)m) = cot beta`
`rArr PQ = (x+2h) cot beta ` metres.
`rArr x(cot alpha - cotbeta) = 2h cot beta rArr x(1/(tanalpha) -(1)/(tanbeta)) = (2h)/(tan beta)`
`rArr x((tanbeta - tan alpha)/(tan alpha + tan beta)) = (2h)/(tan beta) rArr x = (2h tan alpha)/((tan beta - tan alpha))`
`:.`height of the cloud from the surface of the lake
`= (x+h) = {(2htan alpha)/((tan beta + tan alpha))+ h} m`
`= (h(tanalpha + tan beta))/((tan beta - tan alpha))` metres.