Question

Write the eccentricity of the ellipse 9x² + 5y² − 18x − 2y − 16 = 0. 

Answer 1

\[9 x^2 + 5 y^2 - 18x - 2y - 16 = 0\]
\[ \Rightarrow 9\left( x^2 - 2x \right) + 5\left( y^2 - \frac{2y}{5} \right) = 16\]
\[ \Rightarrow 9\left( x^2 - 2x + 1 \right) + 5\left( y^2 - \frac{2y}{5} + \frac{1}{25} \right) = 16 + 9 + \frac{1}{5}\]
\[ \Rightarrow 9 \left( x - 1 \right)^2 + 5 \left( y - \frac{1}{5} \right)^2 = \frac{126}{5}\]
\[ \Rightarrow \frac{\left( x - 1 \right)^2}{\frac{126}{45}} + \frac{\left( y - \frac{1}{5} \right)^2}{\frac{126}{25}} = 1\]
\[ \Rightarrow a^2 = \frac{126}{45} \text{ and } b^2 = \frac{126}{25}\]
\[\text{ Clearly }, a < b\]
\[\text{ Now }, e = \sqrt{1 - \frac{a^2}{b^2}}\]
\[ \Rightarrow e = \sqrt{1 - \frac{\frac{126}{45}}{\frac{126}{25}}}\]
\[ \Rightarrow e = \sqrt{1 - \frac{5}{9}}\]
\[ \Rightarrow e = \frac{2}{3}\]