LocusOfFocusPoint X projects onto line AB in point X', and point F is the reflection of X' in X. The parabola having F as focus and AB as directrix is the locus of point P.
As X goes around a circle, the locus of the focus is an ellipse.
This configuration is of particular interest since the parabola is a function of X (quite different from "y = x2", because here, as X varies, the whole parabola varies).
For more, see GEOMETRY IN ACTION, Chapter 2.