Relative position of a parabola or a hyperbola and an ellipse without computing intersection points

Efficient methods to determine the relative position of two conics are of great interest for applications in robotics, computer animation, CAGD, computational physics, and other areas. We present a method to obtain the relative position of a parabola or a hyperbola, and a coplanar ellipse, directly from the coefficients of their implicit equations, even if they are not given in canonical form, and avoiding the computation of the corresponding intersection points (and their characteristics).