a.math locus

Let M be the mid-point of PQ and its coordinates be (x, y).
Then, for the coordinates of P and Q, we have:
x + k = x2 (Equating the y of the curves together)
x2 - x - k = 0 ... (*)
Then, let x1 and x2 be the coordinates of P and Q, we have:
x = (x1 + x2)/2 as M is the mid-point of PQ.
So x is the half of the sum of roots of (*) which is 1/2 according to the formula of the sum of roots of quadratic equation.
Finally, the locus of M is x = 1/2.