Let a and b be positive real numbers. In the Cartesian coordinate plane, the region between the curve
$$y = a x ^ { 2 } + b$$
and the lines $x = 0$, $x = 2$ and $y = 0$ is divided by the line passing through points $(2,0)$ and $(0, b)$ into two regions whose areas have a ratio of 3.
Accordingly, what is the ratio $\frac { \mathbf { a } } { \mathbf { b } }$?
A) $\frac { 1 } { 2 }$
B) $\frac { 2 } { 3 }$
C) $\frac { 3 } { 4 }$
D) $\frac { 4 } { 5 }$
E) $\frac { 5 } { 6 }$