Official solution
\begin{gathered} \frac{(x-6{)}^{2}}{{a}^{2}}-\frac{(y-2{)}^{2}}{{b}^{2}}=1 \\ \Rightarrow {b}^{2}{x}^{2}-{a}^{2}{y}^{2}-12x{b}^{2}+4y{a}^{2}+36{b}^{2}-4{a}^{2}-{a}^{2}{b}^{2}=0 \\ \frac{{b}^{2}}{{a}^{2}}=3\Rightarrow {e}^{2}=1+\frac{{b}^{2}}{{a}^{2}} \\ \Rightarrow {e}^{2}=4 \\ \Rightarrow e=2
2ae=4 \\ \Rightarrow a=1\Rightarrow b=\sqrt{3} \\ \frac{(x-6{)}^{2}}{1}-\frac{(y-2{)}^{2}}{3}=1 \\ \Rightarrow 3{x}^{2}-{y}^{2}-36x+4y+108-4-3=0 \\ \Rightarrow 3{x}^{2}-{y}^{2}-36x+4y+101=0 \\ \Rightarrow \alpha =36,\beta =4,\gamma =101 \\ \Rightarrow \alpha +\beta +\gamma =141 \end{gathered}