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Question

# If 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 be β and the distance of the cloud from the point of observation is x. Find x if α=30o,β=45o and h=250m

A
1369m
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B
1672m
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C
1428m
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D
1267m
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Solution

## The correct option is A 1369mLet a be a point h metres above the lake AF and B be the position of the cloud.Draw a line parallel to EF from A on BD at C.But,BF=DFLet, BC=mso, BF=(m+h)⇒BF=DF=(m+h)Consider △BAC,AB=mcosecα................(1)and, AC=mcotαConsider △ACD,AC=(2h+m)cotβTherefore, mcotα=(2h+m)cotβ⇒m=2hcotβ(cotα−cotβ)Substituting the value of m in (1) we get,AB=cosecα[2hcotβ/(cotα−cotβ)]=2hsecα(tanβ−tanα)Now, Putting given values in above Equationwe get, 2×250×2√31−1√3=1000√3−1=1000.73=1369Therefore, Correct Answer is 1369

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