Let $a , b , c$ be three non-zero real numbers such that the equation
$$\sqrt { 3 } a \cos x + 2 b \sin x = c , x \in \left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right]$$
has two distinct real roots $\alpha$ and $\beta$ with $\alpha + \beta = \frac { \pi } { 3 }$. Then, the value of $\frac { b } { a }$ is $\_\_\_\_$.