Given a continuous, differentiable curve, find the mid point and radius of a circle that bisects the x-axis and is tangential to the curve at a specific point on the curve.

It looks something like the blue circle in this diagram. These are Mohr's circles.

We know the point where the circle touches the curve, we know the gradient at that point, we know that the circle must be perpendicular to the x axis at y = 0. I think that's sufficient information. But how to calculate it? Geometrically or, more interesting, analytically?

(For the actual work, the students were told that trial and error was acceptable - but I'm interested in how to solve it exactly.