Elements of an ellipse by completing the square

Text

Given $9x^2 + 4y^2 - 36x + 8y + 4 = 0$, complete the square to reduce it to canonical form and find: type of conic, centre, semi-axes, foci and eccentricity.

Solution

$9(x-2)^2+4(y+1)^2=36$, i.e. $\frac{(x-2)^2}{4}+\frac{(y+1)^2}{9}=1.$ It is an ellipse with centre $C(2;-1)$, $a=2$ (horizontal semi-axis) and $b=3$ (vertical, the larger). Since the major axis is vertical, $c=\sqrt{9-4}=\sqrt5$ and the foci are $F\left(2;-1\pm\sqrt5\right)$. The eccentricity is $e=\frac{c}{b}=\frac{\sqrt5}{3}\approx 0{,}745.$

$\boxed{C(2;-1),\ a=2,\ b=3,\ F(2;-1\pm\sqrt5),\ e=\tfrac{\sqrt5}{3}}$