A round baloon of radius r subtends an angle 'a' at the eye of the observer while the angle of elevation of the two aeroplanes from the height of the centre is 'b'. Prove that the height of the centre of the baloon is r sin b cosec a/2.

Dear student,

Let _ be the centre of the ballon of radius ‘r’ and ‘p’ the eye of the observer. Let

PA, PB be tangents from P to ballong. Then ÐAPB = _ .

\ÐAPO = ÐBPO =

2

q

Let OL be perpendicular from O on the horizontal PX. We are given that the

angle of the elevation of the centre of the ballon is f i.e.,

ÐOPL = f

In DOAP, we have sin

2

q

=

OP

OA

_ sin

2

q

=

OP

a

OP = a cosec

2

q

In DOP L, we have sinf =

OP

OL

_ OL = OP sin f = a cosec

2

f

sin _.

Hence, the height of the center of the balloon is a sin _ cosec _ /2.