Q. Let S and S' be the foci of the ellipse and B be any one of the extremities of its minor axis. If 'S'BS is a right angled triangle with right angle at B and area ($\Delta$S'BS) = 8 sq. units, then the length of a latus rectum of the ellipse is :
Solution:
$m_{SB} . m_{SB} \, = \, -1$
$b^2 \, = \, a^2 e^2 \, \, \, \, \, ...(i)$
$\frac{1}{2}S'B . SB = 8$
$S'B. SB = 16$
$a^2c^2 + b^2 \, = \, 16 \, .........(ii)$
$b^2 \, \, = \, \, a^2 \, \, \, (1 - e^2) \, .......(iii)$
using (i),(ii), (iii) a = 4
$ \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, b = 2\sqrt{2}$
$ \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, e = \frac{1}{\sqrt{2}}$
$\therefore \, \, \ell (L.R) \, \, = \frac{2b^2}{a} \, = \, 4$
