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