Question

What's the physical difference between the parabolic and the hyperbolic mirror? I know that the parabolic one can collimate light rays coming from a point source at the focus point. How does the hyperbolic mirror differ from that?

Answer 1

A parabola is a locus of all points which have the same displacement from a given point as they have perpendicular displacement from a given line.
A hyperbola is a locus of all points which have a constant difference in displacements from two given points:
From this, you can geometrically derive the optical properties of each.