Two Chords, One Fixed Point

Fix the parabola $y^2 = 2px$ and points $A(a, b)$, $B(-a, 0)$ with $ab \ne 0$ and $b^2 \ne 2pa$. For a variable point $M$ on the parabola, the lines $AM$ and $BM$ meet the parabola again at $M_1$ and $M_2$ respectively. Prove that, whenever $M_1 \ne M_2$ exist, the line $M_1M_2$ passes through a fixed point, and find that point.