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

1 Answer
Mar 8, 2016

Ellipse

Explanation:

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

The equation can be reorganized as
(x+2)^2/9 +(y-1)^2/25= 1.

Center C of the ellipse is (-2,1).
Semi axes a = 5 and b = 3.
Major axis is x=-2 is parallel to y-axis.
Eccentricity e = sqrt(9^2-5^2)/5=2sqrt14/5.
For foci S and S', CS = CS' = ae = sqrt14.
Foci: (-2, 1+sqrt14) and (-2,1 -sqrt14)