Putnam 1942 A6

Any circle in the $xy$ -plane is "represented" by a point on the vertical line through the center of the circle and at a distance "above" the plane of the circle equal to the radius of the circle. Show that the locus of the representations of all the circles which cut a fixed circle at a constant angle is a portion of a one-sheeted hyperboloid. By consideration of a suitable family of circles in the plane, demonstrate the existence of two families of rulings on the hyperboloid.