Question

If the length of the major axis of the ellipse \[\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\] is three times the length of minor axis, then its eccentricity is______.

Options

• \[\frac{1}{3}\]

• \[\frac{1}{\sqrt{3}}\]

• \[\sqrt{\frac{2}{3}}\]

• \[\frac{2\sqrt{2}}{3}\]

Answer 1

If the length of the major axis of the ellipse \[\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\] is three times the length of minor axis, then its eccentricity is \[\frac{2\sqrt{2}}{3}\].

Explanation:

\[\because2a=3\cdot2\cdot b\quad\Rightarrow\quad a=3b\]

\[\therefore\quad l=\sqrt{\frac{a^2-b^2}{a^2}}=\sqrt{\frac{\left(3b\right)^2-b^2}{\left(3b\right)^2}}\]

\[=\sqrt{\frac{8b^2}{9b^2}}=\frac{2\sqrt{2}}{3}\]