The locus of the mid-point of the line segment joining the focus of the parabola  y2=4ax to a moving point of the parabola, is another parabola whose directrix is:

Let M(h,k) be the midpoint of the line segment joining the focus S(a,0) and any point Pat2,2at.  Then                                                   h=at2+a2,k=2at+02                                                   ⇒t2=2h−aa & t=ka                                                       ⇒k2a2=2h−aa∴ Locus of (h, k) is y2=a2x−a   ⇒ y2=2ax−a2Its directrix is x−a2=−a2⇒x=0