Question

write an equation for the ellipse with a center at (0,0), a focus at (3,0), and a vertex at (-5,0)

Answer 1

what is a focus again

Answer 2

By symmetry, you can deduce the other forcus and the other vertex. Your h and k are given. Because vertices are (h-a, k) and (h+a, k) you know a, similarly from foci you know c. You can deduce b from\[c =\sqrt{a ^{2}+b ^{2}}\]
You have everything you need for your equation.