Hello, jessismith
Is it possible for the tangents to a parabola at two distinct zeroes
to meet at right angles?
Is it possible to find such a parabola for which the x and y intercepts
and the point of intersection of the two tangents lie on the integer lattice?
What can you say about cubics? Here's a bit of parabolic trivia . . .
The endpoints of a focal chord (a chord through the focus) have tangents
. . which are perpendicular and intersect on the directrix.
So if we place the focus at the origin . . . Code:

*  *

* A F B *
o+o
\*  */
\ * /
\  /
\/
    o     
C
Then the intercepts at $\displaystyle A$ and $\displaystyle B$ have tangents so that $\displaystyle AC \perp BC.$