Question

The eccentricity of the ellipse 5x² + 9y² = 1 is

Options

• 2/3

• 3/4

•  4/5

• 1/2

Answer 1

\[ \frac{2}{3}\]
\[5 x^2 + 9 y^2 = 1\]
\[ \Rightarrow \frac{x^2}{\frac{1}{5}} + \frac{y^2}{\frac{1}{9}} = 1\]
Comparing with the standard equation of the ellipse, we get: 
\[ a^2 = \frac{1}{5}\text{ and }b^2 = \frac{1}{9}, \text{i . e . }a = \frac{1}{\sqrt{5}}\text{ and }b = \frac{1}{3}\]
Here, a > b 
\[\text{ Now, }e = \sqrt{1 - \frac{b^2}{a^2}}\]
\[ \Rightarrow e = \sqrt{1 - \frac{\frac{1}{9}}{\frac{1}{5}}}\]
\[ \Rightarrow e = \sqrt{1 - \frac{5}{9}}\]
\[ \Rightarrow e = \sqrt{\frac{4}{9}}\]
\[ \Rightarrow e = \frac{2}{3}\]