Question

Write the eccentricity of the ellipse $9x^2+5y^2-18x-2y-16=0$

Answer 1

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