$\begin{aligned}
& x=3\left(\frac{\tan \theta+\sqrt{3}}{1-\sqrt{3} \tan \theta}\right) \
& x-\sqrt{3} \tan \theta=3 \tan \theta+3 \sqrt{3} \
& \tan \theta=\frac{x-3 \sqrt{3}}{3+\sqrt{3} x} \ldots(1) \
& 2\left(\frac{\tan \theta+\frac{1}{\sqrt{3}}}{1-\frac{\tan \theta}{\sqrt{3}}}=y\right) \
& 2(\sqrt{3} \tan \theta+1)=y(\sqrt{3}-\tan \theta) \ldots .
\end{aligned}$
using (1) and (2)
$\begin{aligned}
& 2\left(\frac{x-3 \sqrt{3}}{\sqrt{3}+x}+1\right)=y\left(\sqrt{3}-\frac{(x-3 \sqrt{3})}{\sqrt{3}(\sqrt{3}+x)}\right) \
& 2 \sqrt{3}(x-3 \sqrt{3}+x+\sqrt{3})=y(3(\sqrt{3}+x)-x+3 \sqrt{3}) \
& 4 \sqrt{3} x-12=y(2 x+6 \sqrt{3}) \
& x y-2 \sqrt{3} x+3 \sqrt{3} y-6=0 \
& \Rightarrow \alpha=-2 \sqrt{3}, \beta=3 \sqrt{3}, \gamma=-6 \
& \alpha^2+\beta^2+\gamma^2=12+27+36=75
\end{aligned}$