Question

A round balloon of radius r sub tends an angle alpha at the eye of the observer while the angle of elevation of its centre is beta. Prove that the height of the centre of the balloon is: r *sin beta*cosec alpha /2

Answer 1

say ∠BAC= α 
∠DAF= β
height= DF
angle of tangent = ∠ABD= 90

In right angle triangle ABD
sin BAD= BD/AD

sin α/2= r/AD

AD= r cosec α/2

sin DAF= DF/AD
sinβ= DF/AD
AD= DF/sinβ

comparing both equations we get
DF= r sinβ cosec α/2