The correct option is C x=0
Let P(h,k) be the mid-point of the line segment joining the focus (a,0) and a general point Q(x,y) on the parabola. Then,
h=x+a2,k=y2⇒x=2h−a,y=2k.
Put these values of x and y in y2=4ax, we get
4k2=4a(2h−a)
⇒4k2=8ah−4q2⇒k2=2ah−a2
So, locus of P(h,k) is y2=2ax−a2
⇒y2=2a(x−a2)
Its directrix is x−a2=−a2⇒x=0