Take two overlapping circles. Examine the locus of the center of the circles tangential to both.
It seems to be an ellipse with foci at the centers of the two circles.
Prove it.

The polar line is the locus of the intersections of tangent lines at the ends of chords of teh parabola through a fixed point.
Turns out to be conceptually important - not just a curiosity.

The locus of the intersections of the tangents at the end of chords of an ellipse which pass through a common point is a straight line.
A bit of a mouthful, but play with the diagram and it will become clear.