The height of the center of the round balloon of radius r, which subtend an angle ∝ at the eye of an observer and the elevation of whose center from the eye is β, is given by?

Let O be the centre of the balloon of radius r which subtend an angle ∝ at the eye of an observer at E .
If EA and EB are the tangents to the ballon,
then ∠ OEA = ∠ OEB = ∝/2
In triangle ΔOAE, Sin ∝/2 = OA/OE
∴ OE = r cosec 1/2 ∝
In ∠OEL, height of the center of the balloon = h = OE sin β = r Cosec ∝/2 Sin β.