Question

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

Answer 1

$$\begin{gathered} 9x^2+5y^2-18x-2y-16=0 \\ \Rightarrow 9\left(x^2-2x\right)+5\left(y^2-\frac{2y}{5}\right)=16 \\ \Rightarrow 9\left(x^2-2x+1\right)+5\left(y^2-\frac{2y}{5}+\frac{1}{25}\right)=16+9+\frac{1}{5} \\ \Rightarrow 9{\left(x-1\right)}^2+5{\left(y-\frac{1}{5}\right)}^2=\frac{126}{5} \\ \Rightarrow \frac{{\left(x-1\right)}^2}{{\displaystyle \frac{126}{45}}}+\frac{{\left(y-{\displaystyle \frac{1}{5}}\right)}^2}{{\displaystyle \frac{126}{25}}}=1 \\ \Rightarrow a^2=\frac{126}{45}\mathrm{and}{b}^2=\frac{126}{25} \\ \mathrm{Clearly},a<b \\ \mathrm{Now},e=\sqrt{1-\frac{a^2{\displaystyle }}{{\displaystyle b^2}}} \\ \Rightarrow e=\sqrt{1-\frac{{\displaystyle \frac{126}{45}}}{{\displaystyle \frac{126}{25}}}} \\ \Rightarrow e=\sqrt{1-\frac{5}{9}} \\ \Rightarrow e=\frac{2}{3} \end{gathered}$$