Home
Class 10
MATHS
If the angle of elevation of a cloud fro...

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)`

Text Solution

Verified by Experts

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.
Promotional Banner

Similar Questions

Explore conceptually related problems

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(tan beta+tan alpha)quad tan beta-tan alpha

If the angle of elevation of the cloud from a point hm above a lake is A and the angle of depression of its reflection in the lake is B prove that the height of the cloud is h(tan B+tan A)tan B-tan A

If the angle of elevation of a cloud from a foot 4 meters above a lake is beta and the angle of depression of its reflection in the lake is alpha then the height is

The angle of elevation of a stationary cloud from a point h meter above a lake is 15^(@) and angle of depression of its reflection in the lake is beta . Prove that height of the cloud above the lake is (h(tanbeta+tanalpha))/(tanbeta-tanalpha) .

The angle of elevation of a cloud from a point h metre above a lake is theta The angle depression of its reflection in the lake is 45^(@) The height of the cloud is

If the angle of elevation of a cloud from a point h metres above lake is alpha and the angle of depression of its reflection in the lake be beta , prove that the distance of the cloud from the point of observation is (2 h sec alpha)/(tan beta - tan alpha)

If the angle of elevation of a cloud from a point 200m above a lake is 30^(@) and the angle of depression of its reflection in the lake is 60^(@) .Then the height of the cloud above the lake is :