Question

locus of point of trisection of the focal chord of the parabola

Answer 1

The question seems to be incomplete, here is the complete question:

The locus of the point of trisection of all the double ordinates of the parabola y2=lx

Answer:

It seems that you think that the focuses of the both parabolas have the same x-coordinate.

The coordinates of the points both on the parabola y2=lx and on x=t are (t,±lt−−√).

Since the double ordinates on x=t are trisected, we get

(t,1⋅lt−−√+2(−lt−−√)1+2) (t,1⋅(−lt−−√)+2⋅lt−−√1+2),

(t,±lt−−√3)

which are on the parabola  y^2=l/3^2x=l/9x