Three points O (0,0), P ( a, a^2), Q (- b, b^2), a >0, b >0, are on the parabola y =x^2. Let S_1 be the area of the region bounded by the line PQ and…

Three points $\displaystyle \mathrm{O}(0,0), \mathrm{P}\left(\mathrm{a}, \mathrm{a}^2\right), \mathrm{Q}\left(-\mathrm{b}, \mathrm{b}^2\right), \mathrm{a}>0, \mathrm{~b}>0$, are on the parabola $\displaystyle \mathrm{y}=x^2$. Let $\displaystyle \mathrm{S}_1$ be the area of the region bounded by the line PQ and the parabola, and $\displaystyle S_2$ be the area of the triangle OPQ. If the minimum value of $\displaystyle \frac{\mathrm{S}_1}{\mathrm{~S}_2}$ is $\displaystyle \frac{\mathrm{m}}{\mathrm{n}}, \operatorname{gcd}(\mathrm{m}, \mathrm{n})=1$, then $\displaystyle \mathrm{m}+\mathrm{n}$ is equal to $\displaystyle \_\_\_\_$.