Let $\mathrm{a}, \mathrm{b}, \mathrm{c}$ 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 $\_\_\_\_$ .