To generalize the cases of points of intersections of a parabola and a circle

Question

Can a circle and a parabola have one intersection point only other than the case when the vertex of the parabola is just touching the circle's maximum or minimum point (y-value)?

(If yes, can anyone please offer an example? Thank you!)

Answer 1

You can place a circle tangent to the parabola at any point on the parabola. Think of the circle as rolling along the parabola, if that helps you visualize.

If two curves are tangent to one another, some people would say that there’s a single point of intersection, and some would say that there are two intersection points that happen to coincide. So, one intersection or two, depending on how you decide to count.

And, of course, a parabola and a circle can have four intersection points, in some cases.

If you want to think algebraically, rather than geometrically, note that the points of intersection correspond to the real roots of a quartic (degree four) polynomial. A quartic can have either zero, two, or four real roots. In the case where there are two real roots, these two roots might be equal, so two intersection points coincide, and we get the tangency situation described above.

There are other possibilities, too. For example, there might be three real roots that are equal (i.e. a triple root) and one that’s different. At the point corresponding to the triple root, there’s an odd form of tangency where the curves actually cross over each other. See this answer for a nice animation of that case.