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

The focus of the parabola y2 = 4ax is S (a, 0), let P (at2 , 2at) be any point on the parabola then coordinates of the mid-point of SP  are given by x=at2+12, y=2at+02Eliminating ‘t’ we get the locus of the mid-point as y2=2ax−a2  or  y2=2a(x−a/2)          (1)which is a parabola of the form Y2=4AX   (2)where Y=y,X=x−a/2 and A=a/2Equation of the directrix of (2) is X=−ASo equation the directrix of (1) is  x−a/2=−a/2⇒x=0