This Demonstration shows that the center of a unit circle tangent to the parabola is at (0, 5/4). In addition, the segment connecting the point and the tangent point makes a 60° angle with the vertical axes of the parabola.

The radius of the tangent circle with center on the axis at is , where is the steepness of the parabola .

Checking the "answer" checkbox gives the radius of the purple circle. When the circles are tangent, the values of the radii are 1, 2, 3, 4, ….

Checking the "ice cream" checkbox shows the three tangent spheres and the ratio of their volumes to the volume of the paraboloid obtained by revolving about the axis.

Reference

[1] J. Stewart, Calculus: Early Transcendentals, 5th ed., Belmont, CA: Brooks/Cole, 2007, Chapter 3.