Q. 513.6( 19 Votes )

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

Answer :

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)


In ∆ADF


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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