How to detect if an ellipse intersects(collides with) a circle

Question

I want to improve a collision system.

Right now I detect if 2 irregular objects collide if their bounding rectangles collide.

I want to obtain the for recta

Answer 1

An ellipse is defined a the set of points whose sum of the distance to point A and the distance to point B is constant e. (A and B are called the foci of the ellipse).

All Points P, whose sum AP + BP is less than e, lie within the ellipse.

A circle is defined as the set of points whose distance to point C is r.

A simple test for intersection of circle and ellipse is following:

Find
P as the intersection of the circle and the line AC and
Q as the intersection of the circle and the line BC.

Circle and ellipse intersect (or the circle lies completely within the ellipse) if
AP + BP <= e   or   AQ + BQ <= e

EDIT:

After the comment of Martin DeMello and adapting my answer accordingly I thought more about the problem and found that the answer (with the 2nd check) still doesn't detect all intersections:

If circle and ellipse are intersecting only very scarcely (just a little more than being tangent) P and Q will not lie within the ellipse:

So the test described above detects collision only if the overlap is "big enough". Maybe it is good enough for your practical purposes, although mathematically it is not perfect.