The chord joining ($at1^2$, 2at1) and ($at2^2$, 2at2) has slope $\frac{2}{t1+t2}$ 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 $\frac{a(t1^2+t2^2}{2}$, a(t1+t2)).