$a + t = ?$

Let $P(x) = x^2 − ax + 8$ with a a positive integer, and suppose that $P$ has two distinct real roots $r$ and $s$. 

Points $(r, 0)$, $(0, s)$, and $(t, t)$ for some positive integer $t$ are selected on the coordinate plane to form a triangle with an area of $2021$. 

Determine the minimum possible value of $a + t$.

Stanford Math Tournament 2021