Question

Area of the quadrilateral formed by joining the extremities of the major axis and minor axis of the ellipse \[ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 \] is \(8\sqrt3\). If the distance between its foci is \(4\sqrt2\), then the eccentricity of the ellipse is

• A. \(\dfrac{1}{\sqrt3}\)
• B. \(\dfrac13\)
• C. \(\dfrac23\)
• D. \(\sqrt{\dfrac23}\)

Hint:

For the ellipse \[ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1, \] the quadrilateral formed by the vertices \((\pm a,0)\) and \((0,\pm b)\) is a rhombus with area \[ 2ab. \] Combine this with \[ c^2=a^2-b^2 \] to find the eccentricity quickly.

Answer 1

The Correct Option is D