  Let $p$ be a prime and $n$ a positive integer such that $\sqrt{\frac p2} + 1
\leq n \leq \sqrt{p}$. For any arithmetic progression $A$ of length $n$ in
$\mathbb{F}_p$, we establish an asymptotic formula for the number of directions
determined by $A \times A \subset \mathbb{F}_p^2$. The key idea is to reduce
the problem to counting the number of solutions to the bilinear Diophantine
equation $ad+bc=p$ in variables $1\le a,b,c,d\le n$; our asymptotic formula for
the number of solutions is of independent interest.
