Theorem: (Hyperbolic Family Radical Axis)

The line containing the centers of an elliptic family of circles is the radical axis of the corresponding hyperbolic family of circles.

Proof:

Let AB be the common chord of the elliptic family of circles. Let E be the midpoint of AB. Since the line through the centers of an elliptic family of circles is the perpendicular bisector of the common chord, we must show the radical axis is the perpendicular line to AB at E. By the Fundamental Radical Axis Theorem, it is enough to show that E has the same power w.r.t. all circles in the hyperbolic family.



Let C be an arbitrary circle in the hyperbolic family. Then C intercepts the circle with diameter AB and center E orthogonally.

Hence, by the Tangent Power Theorem, the power of E w.r.t. C is r² where r is the radius of the circle of diameter AB and center E.

Thus the power of E w.r.t. every circle in the hyperbolic family is r².

Q.E.D.