If two perpendicular straight lines through the focus of the parabola $y^2 = 4ax$ meet its directrix in $T $ and $T'$ respectively. Show that the tangents to the parabola parallel to the perpendicular lines intersect in the mid point of $T T'$.
Progress. Let the points on parabola be $(at^2, 2at)$ and $(ak^2, 2ak)$ then the straight line through focus and these points meet directix at $(-a , -4at/(t^2 -1)]$ and $[-a ,-4ak/(k^2 -1)]$. Please give me a hint on how to continue.