Finding characteristic features of a parabola known by two points and their tangents This is from a math competition so it must not be something really long 
If a parabola touches the lines $y=x$ and $y=-x$ at $A(3,3) $ and $b(1,-1)$ respectively, then 
(A) equation of  axis of parabola is $2x+y=0$ 
(B)slope of tangent at vertex is $1/2$
(C) Focus is $(6/5,-3/5)$
(D) Directrix passes through $(1,-2)$ 
I thought the axis would be the angle bisector of the tangents passing through the focus but it turns out that is not the case in a parabola so how can I find anything..

This figure illustrate the situation.

Using the fact that:

Tangents drawn at the endpoints of a focal chord of a parabola intersect at right angles (on the directrix).

we can say that the focal chord has equation: $y=2x-3$, so we can test that the answer (C): $F=(6/5,-3/5)$ is correct. So we have the focus.
We can also use the fact that:

The tangent at any point of the parabola is equally inclined to the focal distance and the axis of the parabola.

In the figure this mens that the two angles in $A$ and $D$ are equals and from this we can find the axis.
Finally, from the focus and the axis we can find the directrix as the line orthogonal to axis that passes thorough the common point of the two tangents : $C$.
If you o this you can verify that also the answer (D) is correct.
You can find the properties of the tangent used in this answer at: http://www.nabla.hr/CS-ParabolaAndLine2.htm.
