Question

A round balloon of radius r subtends an angle $\alpha$ at the eye of the observer, while the angle of elevation of its centre is $\beta$. The height of the centre of the balloon is____

Answer 1

If the observer is at P and PA, PB are tangents drawn from P to the balloon, then $\angle APB=\alpha$.
$\angle APO=\angle BPO=\alpha /2$.
Further we are given that the angle of elevation of the centre, i.e., $\angle OPQ=\beta$. We have to find the height OQ, of the centre O.
$\frac{OA}{OP}=\sin \frac{\alpha }{2}$ $\therefore OP=r\text{cosec}\left(\frac{\alpha }{2}\right)$ ...(1)
Also, $\frac{OQ}{OP}=\sin \beta$ $\therefore OQ=OP\sin \beta$
= $r\text{cosec}\left(\frac{\alpha }{2}\right)\sin \beta$, by (1)