If for $\theta \in \left[ { - \frac{\pi }{3},0} \right]$, the points $(x,y) = \left( {3\tan \left( {\theta + \frac{\pi }{3}} \right),2\tan \left( {\theta + \frac{\pi }{6}} \right)} \right)$ lie on $xy + \alpha x + \beta y + \gamma = 0$, then ${\alpha ^2} + {\beta ^2} + {\gamma ^2}$ is equal to