What A Lucky Coincidence

Both of these parabolas are symmetric with respect to the coordinate axes. At $y=0$ , we have $x = \pm \sqrt{\frac{a}{b}}$ . The area between these curves is thus partitioned into four equal quadrants and can be computed as:

$A = 4\int_{0}^{\sqrt{a/b}} -x^2 + \frac{a}{b} dx \Rightarrow 4[-\frac{x^3}{3} + \frac{a}{b} x]|_{0}^{\sqrt{a/b}} = \frac{8}{3} (\frac{a}{b})^{3/2} = \frac{a}{b} \Rightarrow \frac{a}{b} = \frac{9}{64} \Rightarrow a+b = \boxed{73}.$