Question

The eccentricity of the ellipse 9x² + 25y² − 18x − 100y − 116 = 0, is

Options

• 25/16

• 4/5

• 16/25

• 5/4

Answer 1

\[\frac{4}{5}\]
\[9 x^2 - 18x + 25 y^2 - 100y - 116 = 0\]
\[ \Rightarrow 9( x^2 - 2x) + 25( y^2 - 4y) = 116\]
\[ \Rightarrow 9( x^2 - 2x + 1) + 25( y^2 - 4y + 4) = 116 + 100 + 9\]
\[ \Rightarrow 9(x - 1 )^2 + 25(y - 2 )^2 = 225\]
\[ \Rightarrow \frac{9(x - 1 )^2}{225} + \frac{25(y - 2 )^2}{225} = 1\]
\[ \Rightarrow \frac{(x - 1 )^2}{25} + \frac{(y - 2 )^2}{9} = 1\]
\[\text{Comparing it with }\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1,\text{ we get: }\]
\[a = 5\text{ and }b = 3\]
Here, a > b, so the major and the minor axes of the ellipse are along the x - axis and y - axis, respectively . 
\[\text{ Now,} e = \sqrt{1 - \frac{b^2}{a^2}}\]
\[ \Rightarrow e = \sqrt{1 - \frac{9}{25}}\]
\[ \Rightarrow e = \sqrt{\frac{16}{25}}\]
\[ \Rightarrow e = \frac{4}{5}\]