The correct option is B y2=2ax+4a2
If normal at A(t1) meets again at C(t3) on the parabola y2=4ax, then
t3=−t1−2t1 …(i)
If normal at B(t2) meets again at C(t3) on the parabola, then
t3=−t2−2t2 …(ii)
From equations (i) and (ii),
−t1−2t1=−t2−2t2⇒t1t2=2 …(iii)
Let (h,k) be the mid point of AB, then
h=a(t21+t22)2
⇒2ha=t21+t22 …(iv)
and k=a(t1+t2)
⇒k2a2=t21+t22+2t1t2
From equations (iii) and (iv), we get
k2a2=2ha+4
Hence, required locus is y2=2ax+4a2