# What conic section is 25x^2 + 100x + 9y^2 – 18y = 116?

Mar 8, 2016

Ellipse

#### Explanation:

If a, b and 2h are the coefficients of the terms in ${x}^{2.} {y}^{2}$and xy, then the second degree equation represents en ellipse parabola or hyperbola according as $a b - {h}^{2}$ >. = or < 0.
Here, $a b - {h}^{2}$ = 225 > 0.

The equation can be reorganized as
${\left(x + 2\right)}^{2} / 9 + {\left(y - 1\right)}^{2} / 25$= 1.

Center C of the ellipse is $\left(- 2 , 1\right)$.
Semi axes a = 5 and b = 3.
Major axis is $x = - 2$ is parallel to y-axis.
Eccentricity e = $\frac{\sqrt{{9}^{2} - {5}^{2}}}{5} = 2 \frac{\sqrt{14}}{5}$.
For foci S and S', CS = CS' = ae = $\sqrt{14}$.
Foci: $\left(- 2 , 1 + \sqrt{14}\right) \mathmr{and} \left(- 2 , 1 - \sqrt{14}\right)$