Question

Area of the quadrilateral formed by the ends of major axis and minor axis of an ellipse is $8\sqrt{3}$. The distance between the foci is $4\sqrt{2}$, then the eccentricity of the ellipse is:

• A. $\frac{1}{\sqrt{3}}$
• B. $\frac{1}{3}$
• C. $\sqrt{\frac{2}{3}}$
• D. $4$

Answer 1

Answer: (C)

$\sqrt{\frac{2}{3}}$

Given : $Areaofquadrilateral=\frac{1}{2}\times a\times b=\frac{8\sqrt{3}}{4}$

$⟹ab=4\sqrt{3}$ ......... $\left(i\right)$

We know that, Distance between foci $=2ae=4\sqrt{2}$

$⟹ae=2\sqrt{2}$ ......... $\left(ii\right)$

and $e^2=1-\frac{b^2}{a^2}$

$⟹\frac{8}{a^2}=1-\frac{48}{a^4}$

$⟹a^4-8a^2-48=0$

$⟹(a^2-12)(a^2+4)=0$

$⟹a^2=12$

$⟹a=2\sqrt{3}$

$e=\frac{2\sqrt{2}}{2\sqrt{3}}$ .......... From $\left(ii\right)$

$\therefore e=\sqrt{\frac{2}{3}}$