Question

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

Answer 1

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 `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)`