2021 AIME II Problem 5

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}.