Hello, riemann!
3.Show that circle described on focal chord of a parabola as diameter touches its directrix.
Let the parabola be: .$\displaystyle x^2 \:=\:4py$
This parabola opens upward; its vertex is at the Origin.
The focus is at $\displaystyle (0,p)$
The directrix is: .$\displaystyle y \:=\:p$
The focal chord (latus rectum) has length $\displaystyle 4p.$
The graph looks like this:
Code:

*  *


*  *
F(0,p)
(2p,p) o    o    o (2p,p)
*  *
*  *
*


    +      y = p


The circle has center $\displaystyle (0,p)$ and radius $\displaystyle 2p$.
You should be able to complete the problem now.