The locus of the mid-points of the chords the parabola x2 = 4py  having slope m is a:

Let (h, k) be the mid point of a chord then its equation is  hx−2p(y−k)=h2−4pk (using T=S) slope of this chord is h2p=m⇒h=2pmLocus of (h, k)  is x = 2pm which is line parallel to y-axis at a distance |2pm| from it.