Number of Intersection points between lower half of an ellipse and a circle

Question

An ellipse has its axes parallel to the coordinate system axes and its major axes is parallel to X-axis.

Meanwhile, there is a circle located at the coordinate system origin, whose radius is smaller than the semi major of the ellipse.

Now a curve is constructed as the lower half of the said ellipse.

How do I find out how many intersection points will exist between the circle and the curve? The general case can be 1 point, 2 points and 3 points (shown in the figures below). Is it possible to have 4 intersection points?

Answer 1

Yes, you can have 4 intersection points. Take for instance four points on the half-ellipse symmetric around the minor axis and the circle passing through them. But non-symmetric solutions are also possible.

Answer 2

Your situation is of intersection between two conics of second order. Let us consider the total number r of points of intersection comprehensively.

Apart from zero number of intersections for non-intersecting complex case, you can only have an even number of real intersection points either two or four, while solving for their solutions.

In tangential case these become correspondingly one or two double points with discriminant vanishing for a double root.

You cannot have odd number of real intersections no matter where you draw them in the $(x,y)$ plane. If you draw your right figure fully you would encounter one more intersection at right in your diagram.