To find the region in question, we want to find $(a, b)$ such that the discriminant of the quadratic is not positive. In other words, we want $$4(a+b-7)^{2}-4(a)(2b) \leq 0 \Leftrightarrow a^{2}+b^{2}-7a-7b+49 \leq 0 \Leftrightarrow(a-7)^{2}+(b-7)^{2} \leq 49$$ which is a circle of radius 7 centered at $(7,7)$ and hence has area $49 \pi$.