Locus of points from an ellipse

A recent episode of WNYC's RadioLab (find it here) offhandedly mentions that the locus of all points a given distance from an ellipse is never an ellipse. My question is, what is that locus? How do you know? Can you write an equation of it?

In the episode, the way the problem is actually posed deals with a swimming pool in the shape of the ellipse. A one-foot concrete rim is placed around the pool- is the outer edge of the concrete an ellipse? They say the answer is 'No, never', and leave it at that.