# If the angle of elevation of a cloud from a point h metres above a lake is a and the angle of depression of its reflection in the lake be b, prove that the distance of the cloud from the point of observation is

In the fig A is the point of observation and C is the position of the cloud.

Let the distance between the cloud and point of observation is x.

In ∆ACD

Sin α =

CD = AC Sin α x Sin α

Cos α =

AD = x Cos α …………………….(1)

CE = CD + DE = (h + x Sin α)

EF = CE = (h + x Sin α)

DF = DE + EF = (h + h + x Sin α) = (2h + x Sin α) ……………..(2)

tan β =

On substituting value of DF & AD

from above eqns (1) and (2)

tan β =

=

2h Cos β + x Sin α Cos β = x Sin Cos α

x (Cos α Sin - Sin α Cos β ) = 2h Cos β

x =

On dividing numerator and denominator by Cos α Cos β, we get

x =

Therefore the distance between cloud and point of observation is m

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