Question

If the length of the major axis of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is $3$ times the length of the minor axis, its eccentricity is

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

Answer 1

The correct option is D

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

Explanation for the correct option:

The equation of the ellipse is $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,\left(a>b\right)$.

The length of the major axis can be given by $2a$.

The length of the major axis can be given by $2b$.

Thus, $2a=3\times 2b$

$\Rightarrow \frac{b}{a}=\frac{1}{3}$.

Therefore the eccentricity of the ellipse can be given by, $e=\sqrt{1-{\left(\frac{b}{a}\right)}^2}$

Put $\frac{b}{a}=\frac{1}{3}$.

$$\begin{gathered} \Rightarrow e=\sqrt{1-{\left(\frac{1}{3}\right)}^2} \\ \Rightarrow e=\sqrt{1-\frac{1}{9}} \\ \Rightarrow e=\sqrt{\frac{9-1}{9}} \\ \Rightarrow e=\sqrt{\frac{8}{9}} \\ \Rightarrow e=\frac{2\sqrt{2}}{3} \end{gathered}$$

Hence, option (D) is the correct answer.