The angle of elevation of a cloud from a point h metres above the surface of a lake is θ and the angle of its reflection in the lake is Φ. Prove that the height of the cloud above the lake is . . .

Question : The angle of elevation of a cloud from a point h metres above the surface of a lake is θ and the angle of its reflection in the lake is Φ. Prove that the height of the cloud above the lake is h[(tanΦ+tanθ)/(tanΦ-tanθ)].

Doubt by Muskan

Solution : 

Let the height of the cloud above the lake be y. 
AB = h
PN = NQ = y
AL = x
PL = y-h
QL = y+h

In Rt. ∆PLA
tanθ = PL/AL
tanθ = (y-h)/x
x = (y-h)/tanθ  — (1)

In Rt. ∆QLA

tanΦ = QL/AL
tanΦ = (y+h)/x
xtanΦ = y+h
[(y-h)tanΦ]/tanθ = y+h [Using (1)]
(y-h) tanΦ = (y+h)tanθ
y tanΦ - htanΦ = y tanθ + htanθ
y tanΦ - y tanθ = htanθ + htanΦ
y(tanΦ - tanΦ) = h (tanθ + tanΦ)
y(tanΦ - tanΦ) = h (tanΦ + tanθ )
y = h [(tanΦ+tanθ)/(tanΦ-tanθ)]

Hence Proved