Question

If the area of the auxillary circle of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b)$ is twice the area of the ellipse, then the eccentricity of the ellipse is______

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

Answer 1

The correct option is D $\frac{\sqrt{3}}{2}$

The radius of the auxiliary circle is half the major axis and so its area would become $\pi \times a^2$

The area of the ellipse is $\pi ab$

According to the given condition, $\pi a^2=2\times \pi ab$

$\Rightarrow a=2b$

$\therefore e=\sqrt{1-\frac{b^2}{a^2}}=\sqrt{1-\frac{b^2}{4b^2}}$

$=\sqrt{1-\frac{1}{4}}$

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