Question

A spherical balloon of radius r subtends an angle α at the eye of an observer, when the angular elevation of its centre is β.The height of the centre of the balloon is

(a) r sin α/2 cosec β

(b) r cosec α sin β/2

(c) r cosec α/2 sin β

(d) r sin α cosec β/2

Answer 1

(c) r cosec α/2 sin β

Let O be the centre of the balloon of radius r and E the position of the eye of the observer.

Let EA, EB be the tangents from E to the balloon.

Then,

∠EAB = α.

Let OL be the perpendicular from O on the horizontal line EX. 

Given, 

∠OEL = β

∠OLE = 90°, 

∠OBE = 90° = ∠OAE (∵ radius \(\perp\) to the tangent at point of contact)

Also,

∠BEO = ∠AEO = \(\frac{\alpha}{2}\)  (∵ DAEO ≅ DBEO)

In rt. Δ OAE,

\(sin\frac{\alpha}{2}=\frac{OA}{OE}\)

⇒ OE = OA cosec \(\frac{\alpha}{2}\) = r cosec \(\frac{\alpha}{2}\)    ....(i)

In rt. Δ OLE,

sin β = \(\frac{OL}{OE}\)

⇒ OL = OE sin β 

= r cosec \(\frac{\alpha}{2}\)sin β   [From (i)]