Finding the ellipse equation from a vector and two foci

[Lurid]

Homework Statement

Given the vector (6,-15) and foci (6,10) (6,-14),
Find the equation of the conic.

Homework Equations

Vector = <6,-15>
(x-h)²/a² + (y-k)²/b² = 1

Foci (h+c,k) and (h-c,k)
vertices (h+a,k), (h-a,k)

The Attempt at a Solution

k = 6
h=-2
a=-12
I'm not sure what I'm doing. How do I know that this conic is an ellipse, and not a hyperbola? And how do I find b and c, after finding a?

edit:
How does a vector even relate to the equation of an ellipse?

 

[vela]

I assume the point (6, -15) is a point on the ellipse.

The expressions you've written down for the locations of the foci are for when the foci lie on the same horizontal line. In this case, the foci lie on the same vertical line.

Your value for a is wrong.