Prove that the locus of the midpoints of the focal chords of the parabola is another parabola whose vertex is the focus of .
Given the parabola is . Let be the midpoint of a focal chord.
Equation of the chord with as midpoint is
Since this equation passes thtough the focus , we have
Therefore, the locus of the midpoint is
This is a parabola with vertex which is the focus of the given parabola.