Edit: I got the answer by finding points of intersection between the line passing through B and the focus and the parabola, but it didn't seem like the best solution. Any other ideas?
The Segments AA' and BB' are focal chords of the parabola $x^2 = 2y$. The coordinates of A and B are $(4,8)$ and $(-2,2)$ respectively.
The question is then split into parts
a) find the equation of the line through A and B'
b) find the equation of the line through B and A'. But I think that those parts are necessary to arrive at the conclusion that
c) (show that) the two lines you found intersect at the directrix.
I found the focus of the parabola, $(0,1/2)$ and the directrix $y=-1/2$. How do I find the coordinates of B' and A' so that I can find the equations of the two lines in the question? Does it have to do with the fact that the segment from B to the focus has the same slope as the segment from the focus to B'?
The coordinates of the focal point are correct, as well as the equation of the directrix. Now you can proceed purely analytically. Hints: first show:
The line through $A$ and $A'$ is $y=1\frac{7}{8}x+\frac{1}{2}$
The line through $B$ and $B'$ is $y=-\frac{3}{4}x+\frac{1}{2}$.
From this (by intersecting with the parabola, which gives you a quadratic equation) deduce that:
$A'=(-\frac{1}{4},\frac{1}{32})$ and $B'= (\frac{1}{2},\frac{1}{8})$.
Now show that:
The line through $A$ and $B'$ is $y=2\frac{1}{4}x-1$
The line through $A'$ and $B$ is $y=-1\frac{1}{8}x-\frac{1}{4}$.
These last lines intersect in $(\frac{2}{9},-\frac{1}{2})$, which is a point on the directrix.