- Consider the Cartesian equation
Prove that this equation represents all conic sections symmetric about the origin with foci at and .
- Let be a fixed constant and let be the set of all such conics as takes on all positive values other than . Prove that every curve in the set satisfies the differential equation
- Prove that the set is self-orthogonal. This means that the set of all orthogonal trajectories of curve in is the set itself.