A balloon of radious r suntends an angle` alpha` at the eyes of an observer and the center of balloon from the eye is `beta` . Find the ofcentre of the centre of the balloon from the eye of observer .

Options

A. `(r sin beta)/(sin ((alpha)/(2)))`
B. `(r sin beta)/(sin ((alpha)/(4)))`
C. `(r sin ((beta)/(2)))/(sin alpha)`
D. ` (r sin alpha)/(sin ((beta)/(2)))`

Answer

A

Explanation

Let 'A' be the position of eye.
Let 'O' be the centre of spherical balloon.
Let 'h' be the height of centre of balloon.

From figure, in `DeltaOAD, sin (alpha)/(2) = (OD)/(OA)=(r)/(OA)`
`rArr OA=(r)/(sin (alpha)/(2)) " " ...(1)`
In `Delta OAB, sin beta=(OB)/(OA)=(h)/(OA)`
`rArr h = OA . sin beta`
`=(r. sin beta)/(sin (alpha)/(2)) ` (from (1))