In the above figure you can see a circumference of radius $r$ and a point $P$ outside it. Assume that you draw a line going through $P$ that hits the circle at some point $C$. When does line $PC$ have the maximum and minimum slope? Is it when it's tangent to the circle? Why? (Point $P$ has to be "under" and "to the left" with respect to the circle).
When I tried to prove it, I thought that it was obvious that the tangent line has more slope than some secant lines (Assume that point P goes clockwise until it gets to the point of tangency). But I do not know what happens after the point of tangency but the secant lines that cut the circle after the point of tangency also cut it before it so then it would be one of those lines that have less slope, but that´s not a rigurous way to prove it.