What are four similarities between hyperbolas and ellipses?

Answer 1

1. both are conic sections ( the intersection of a cone by a plane)
2. both have two directrices
3. both have two foci
4. the ellipse is the locus of points so that sum of the distances to the foci is constant/the hyperbola is the locus of points so that difference of the distances to the foci is constant.
5. In projective geometry they are equivalent. Hyperbolas are just ellipses that cross over infinity. (imagine watching a big giant ellipse that goes off in one direction past the horizon line and then reemerges from the horizon line behind you. It would look just like a hyperbola.)
6. The hyperbola is an ellipse with an imaginary semi-minor axis (a negative quare root; hence the change in sign in the hyperbola's formula)