Question

The area of an ellipse is $8\pi$ sq. units. Its distance between the foci is $4\sqrt{3}$, then $e=$

• A. $\sin {30}^0$
• B. $\sin {45}^0$
• C. $\sin {60}^0$
• D. $\sin {75}^0$

Answer 1

The correct option is C $\sin {60}^0$

$\pi ab=8\pi$

$ab=8$

$2ae=4\sqrt{3}$

$ae=2\sqrt{3}$

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

${\left(\frac{2\sqrt{3}}{a}\right)}^2=1-{\left(\frac{8}{a^2}\right)}^2$

$\frac{12}{a^2}=1-\frac{64}{a^4}$

$\frac{12}{a^2}=\frac{a^4-64}{a^4}$

$12a^2=a^4-64$

$a^2=16$

$a=\pm 4$

$e=\frac{2\sqrt{3}}{4}$

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

$=\sin {60}^{\circ }$