I was solving a problem that asked for the equation of the ellipse with the following properties: vertex at $(-10,5)$, focus at $(-2,5)$, eccentricity $\frac{1}{2}$. I think I found two such ellipses, but since the book only lists one. I want to make sure I'm not mistaken.
The larger, which the book lists, is $$\frac{(x-6)^2}{256}+\frac{(y-5)^2}{192}=1$$ and the smaller is $$\frac{(x+\frac{14}{3})^2}{256/9}+\frac{(y-5)^2}{192/9}=1$$
Both have $(-10,5)$ as their leftmost vertex, and the large ellipse has $(-2,5)$ as it's leftmost focus. As far as I can tell, the smaller is a scaled-down version of the larger, preserving eccentricity $\frac{1}{2}$, with $(-2,5)$ as its rightmost focus. Does that all check out?