Question

A round balloon of radius 'r' subtends an angle alpha at the eye of the observer while the angle of elevation of its center is P. Prove that the height of the centre of the balloon is r sin beta cosec a/2

Answer 1

let o be the center of the balloon of radius r and P the eye of the observer.
Let PA, pb the tangent from P to the balloon.
then angle APB=α

ANGLE APLO=ANGLE BPO=α divide by 2

let OL be a perpendicular From O on the horizontal PX.
we are given that the angle of elevation of the centre of the balloon is β I. e, angle OPL=β

in triangle OAP we have

sin α/2=OA/OP

sin α/2=r/OP

OP=r cosec α/2

in triangle OPL we have

sin β=OL/OP

using equation 1
OL=OP sin β=r cosec α/2*sinβ