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Question

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)(tanβtanα) metres

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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 PQCD. Then,

QPC=α,QPD=β,

BQ=AP=h metres

Let CQ=x metres. Then,

BD=BC=(x+h) metres

From right ΔPQC, we have

PQCQ=cot αPQx m=cot α

PQ=x cotα metres ........ (i)

From right ΔPQD, we have

PQQD=cot βPQ(x+2h)m=cot β

PQ=(x+2h) cotβ metres ........ (ii)

From (i) and (ii), we get

xcotα=(x+2h)cotβ

x(cotαcotβ)=2hcotβx(1tanα1tanβ)=2htanβ

x(tanβtanαtanαtanβ)=2htanβx=2h tanα(tanβtanα)

height of the cloud from the surface of the lake

=(x+h)={2htanα(tanβtanα)+h}m

=h(tanα+tanβ)(tanβtanα) metres

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