Prove that for a parabola of the form x2 = 4py, for ANY focal chord, lines passing through either endpoint of the focal chord and the vertex will pass through the directrix at the foot of the perpendicular line that can be dropped from the opposite focal chord endpoint to the directrix.
Your proof should use variables to show that it holds for any choice of focal chord.
I know that for any focal chord of any length, one endpoint's coordinates are (x,y) and (-4p^2/x,p^2,y). But how can I use that to solve this question (assuming I even need it)?
Basically, I need to prove that the red and blue line both hit the directrix at the same point, the blue line passes through one endpoint of the (green) focal chord and the vertex, the red line is perpendicular to the green focal chord.