For positive real numbers $s$, let $\tau (s)$ denote the set of all obtuse triangles that have area $s$ and two sides with lengths $4$ and $10$. The set of all $s$ for which $\tau (s)$ is nonempty, but all triangles in $\tau (s)$ are congruent, is an interval $[a,b)$. Find $a^2+b^2$.