The locus of the point of intersection of two normals to a parabola which are at right angles to one another is

The equation of the normal to the parabola y2=4ax is y=mx−2am−am3 It passes through the point (h,k) if k=mh−2am−am3 or  am3+m(2a−h)+k=0-----(1) Let the roots of the above equation be m1,m2, and m3. Let the perpendicular normals correspond to the values of m1 and m2 so that m1m2=−1 .  From (1), m1m2m3=−k/a. Since m1m2=−1,m3=k/a .  Since m3 is a root of (i), we have aka3+ka(2a−h)+k=0 or  k2+a(2a−h)+a2=0 or k2=a(h−3a) Hence, the locus of (h,k) is y2=a(x−3a)