The study of the power of a point with respect to a circle and its application to orthogonal circles, bundles of circles, etc., has received a lot of attention in the past. In this paper, we show how the concept of conjugate ellipses generalizes the concept of orthogonal circles. It is also shown that it is possible to design 3R serial regional robots whose inverse kinematics can be reduced to the computation of the intersection between two conjugate ellipses which, in turn, can be reduced to the intersection of an ellipse and a line by relying on the concept of radical conic. The relevance of these findings is illustrated through an example.