The chord joining (at1^2, 2at1) and (at2^2, 2at2) has slope 2/(t1+t2) and equation (t1+t2)y = 2x + 2at1t2. For a focal chord, t1t2=-1, so the equation becomes (t1+t2)y = 2x - 2a. The midpoint of the chord at parameters t1, t2 is (a(t1^2+t2^2)/2, a(t1+t2)).