Q:

Intersection of circle and ellipse

I'm looking for the points of intersection of a circle
$x^2 + y^2 = r^2$ ($r$ is known, origin is $(0,0)$)
and an ellipse
$(x - x_0)^2 / a^2 + (y-y_0)^2 / b^2 = 1$ ($a,b,x_0,y_0$ are known).
Actually i do only need the Angles $\varphi$ at which the circle with radius $r$ is intersecting the ellipse.

A:

One method is to combine the two quadratics in $x$ and $y$ into a single quartic in just $x$.
Expand the second equation, and get $Ax^2+By^2+Cx+Dy=E$.
Substitute $y^2=r^2-x^2$, and rearrange to get $y=Fx^2+Gx+H$.
Substitute that back into the first equation to get a quartic polynomial in $x$.

