A and b are on the parabola y2=2x.

y 1^2=2x 1

Y2 2 = 2x2 subtraction

(y2-y 1)(y2+y 1)= 2(x2-x 1)( 1)

AB midpoint x = [x1+x2]/2y = [y1+y2]/2.

k =(y- 1)/(x- 1)=(y2-y 1)/(x2-x 1)

Substitute (1) to get

[(y- 1)/(x- 1)]*2y=2

y(y- 1)=x- 1

The locus equation of the midpoint of the chord AB is y 2-y = x-1.