Question

If the major axis of an ellipse is three times the minor axis, then its eccentricity is equal to

Options

• \[\frac{1}{3}\]

   

• \[\frac{1}{\sqrt{3}}\]

   

• \[\frac{1}{\sqrt{2}}\]

   

• \[\frac{2\sqrt{2}}{3}\]

   

• \[\frac{2}{3\sqrt{2}}\]

   

Answer 1

\[\frac{2\sqrt{2}}{3}\]
Length of the major axis = 2b
Length of the minor axis = 2a
According to question, the major axis of the ellipse is three times the minor axis.
\[i . e . 2b = 3(2a)\]
\[ \Rightarrow 2b = 6a\]
\[ \Rightarrow a = 2b/6\]
\[ \Rightarrow a = b/3, b = 3a\]
Here, a < b, so the major and the minor axes of the ellipse are along the x - axis and the y - axis, respectively.
\[\text{ Now, }e = \sqrt{1 - \frac{a^{{}^2}}{b^2}}\]
\[ \Rightarrow e = \sqrt{1 - \frac{\frac{b^2}{9}}{b^2}}\]
\[ \Rightarrow e = \sqrt{1 - \frac{1}{9}}\]
\[ \Rightarrow e = \sqrt{\frac{8}{9}}\]
\[ \Rightarrow e = \frac{2\sqrt{2}}{3}\]