jmc

Let $F_1=(3,-4)$ and $F_2=(-5,8)$. Then, given a point $P=(x,y)$, we can rewrite the given equation as $$P{F}_1+P{F}_2=20$$by the distance formula. Therefore, the ellipse has foci $F_1$ and $F_2$, and so the answer is $$F_1{F}_2=\sqrt{(3+5{)}^2+(-4-8{)}^2}=\sqrt{8^2+12^2}=\boxed{4\sqrt{13}}.$$