Consider an ellipse, whose center is at the origin and its major axis is along the $x$-axis. If its eccentricity is $\frac{3}{5}$ and the distance between its foci is $6$, then the area (in sq. units) of the quadrilateral inscribed in the ellipse, with the vertices as the vertices of the ellipse, is:

• A

  $32$
• B

  $80$
• Correct answer
  C

  $40$
• D

  $8$

Solution

Given eccentricity of the ellipse $$$e=\frac{}{}\frac{3}{5}.$

Distance between foci  is  $2ae=6$
$\Rightarrow a=5.$

Also, ${}^{}$$b^2{}^{}=a^2{}^{}(1-e^2)$

$\Rightarrow$$$ $b^2=25\left(1-\frac{9}{25}\right)$

$\Rightarrow$ $b=4$

Area $=4\left(\frac{1}{2}ab\right)=2ab=40$$$

Hence, the correct answer is option: 'C'.