The angle of elevation of a cloud from a point h metres above a lake is α and the angle of depression of its reflection in the lake is β. Prove that the height of the cloud is h(tanβ+tanα)(tanβ−tanα)metres
Open in App
Solution
Let AB be the surface of the lake and 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,
∠QPC=α,∠QPD=β,
BQ=AP=hmetres
Let CQ=xmetres. Then,
BD=BC=(x+h)metres
From right ΔPQC, we have
PQCQ=cotα⇒PQxm=cotα
⇒PQ=xcotαmetres ........ (i)
From right ΔPQD, we have
PQQD=cotβ⇒PQ(x+2h)m=cotβ
⇒PQ=(x+2h)cotβmetres ........ (ii)
From (i) and (ii), we get
xcotα=(x+2h)cotβ
⇒x(cotα−cotβ)=2hcotβ⇒x(1tanα−1tanβ)=2htanβ
⇒x(tanβ−tanαtanαtanβ)=2htanβ⇒x=2htanα(tanβ−tanα)
∴ height of the cloud from the surface of the lake