Intersection of circle and ellipse

Question

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.

Answer 1

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 $x$.

Answer 2

Hint:$$x^2 + y^2 = r^2\implies x^2 + y^2 - r^2=0$$ and $$(x - x_0)^2 / a^2 + (y-y_0)^2 / b^2 = 1\implies (x - x_0)^2 / a^2 + (y-y_0)^2 / b^2 - 1=0$$

So you're solving $$x^2 + y^2 - r^2=(x - x_0)^2 / a^2 + (y-y_0)^2 / b^2 - 1$$

But the solution will be long, however if $x_0=y_0=0$ then $$y=\pm\frac{\sqrt{a^2r^2+a^2x^2+a^2-x^2}}{\sqrt{\frac{a^2}{b^2}-a^2}}$$