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)
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