The straight line drawn from a point P on a parabola to the vertex cuts the directrix at T. Prove that if the focus is at S, that TS is parallel to the tangent at P.

S = (2as, as^2)

P = (2ap, ap^2)

T = (2at, at^2)

gradient PT = (at^2 - ap^2)/2at - 2ap

= a (t^2 - p^2)/ 2a(t-p)

= (t + p)/2

gradient TS = (at^2 - as^2)/(2at - 2as)

= (t + s)/2

I can't seem to get gradient PT = gradient TS

Help please

Thanks in advance