<s>[INST] 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.
 [/INST] 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 $$y=\pm\frac{\sqrt{a^2r^2+a^2x^2+a^2-x^2}}{\sqrt{\frac{a^2}{b^2}-a^2}}$$
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