In the case of a arbitrary circle overlapping with an ellipse in the origin. I don't think there is a closed expression to find the intersection points. However, I was wondering, is there a (well?) known closed expression (this may be a series expansion) to describe the arc length (red) of the part of the circle overlapping with an ellipse?
Let us define the ellipse as
$$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$
and the circle
$${(x-x_0)^2}+ {(y-y_0)^2} = r^2$$
I think the polar coordinates will be perhaps more helpful for this problem :
$$\begin{array}{lcl}x&=&r\cos(t) + x_0\\y&=&r\sin(t) + y_0\end{array}$$