Let P and Q be points on the parabola, with coordinates (2ap, ap^2) and (2aq, aq^2) respectively.
Equation of chord is:
y - ap^2 = (aq^2 - ap^2)(x - 2ap)/(2aq - 2ap)
y - ap^2 = a(q - p)(q + p)(x - 2ap)/2a(q - p)
y - ap^2 = (q + p)(x - 2ap)/2
Since this is a focal chord, (a, 0)...