This problem requires us to combine concepts from circles and ellipses, specifically finding the center and radius of a circle, determining the equation and properties of an ellipse, and calculating focal distances.

1. Key Concepts and Formulas

• Circle Equation: The general equation of a circle is $x^2 + y^2 + 2gx + 2fy + c = 0$. Its center is $C(-g, -f)$ and its radius is $r = \sqrt{g^2+f^2-c}$. Alternatively, the standard form is $(x-h)^2 + (y-k)^2 = r^2$, where $C(h,k)$ is the center and $r$ is the radius.