It is known that the eccentricity of a hyperbola is >1, and that of a circle is 0. These are both conic sections of (the lower nappe of) a (double) cone, with r the radius and h the height of the cone, and theta the angle of the intersecting plane to the vertical axis of the cone:
Assume the circle is a horizontal section of a plane with this theoretical cone, and the hyperbola a vertical section of a plane with the same cone. So these 2 planes are perpendicular, and assume these 2 planes intersect right in the point where the circle "touches" the outside of the cone (intersecting conics). In other words: the maximum of the (bottom part) of the vertical hyperbola touches the "left" side of the circle in 1 point. So it is a vertical hyperbola: both foci and midpoint lie on a vertical line (parallel to the Y-axis in a 2-dimensional plane).
Quick version: horizontal circle with unknown radius touches vertical hyperbola with known equation in a 3-dimensional space: can I know the radius of the circle using conic sections and eccentricity? Or some other way?