Question

The eccentricity of the ellipse 4x² + 9y² = 36 is

Options

• \[\frac{1}{2\sqrt{3}}\]

   

• \[\frac{1}{\sqrt{3}}\]

   

• \[\frac{\sqrt{5}}{3}\]

   

• \[\frac{\sqrt{5}}{6}\]

   

Answer 1

\[\frac{\sqrt{5}}{3}\]
\[4 x^2 + 9 y^2 = 36\]
\[ \Rightarrow \frac{x^2}{9} + \frac{y^2}{4} = 1\]
\[\text{ Comparing it with }\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \text{ we get: }\]
\[a = 3\text{ and }b = 2\]
Here, a > b, so the major and the minor axes of the ellipse are along the x - axis and y - axis, respectively.
\[\text{ Now }, e = \sqrt{1 - \frac{b^2}{a^2}}\]
\[ \Rightarrow e = \sqrt{1 - \frac{4}{9}}\]
\[ \Rightarrow e = \sqrt{\frac{5}{9}}\]
\[ \Rightarrow e = \frac{\sqrt{5}}{3}\]