If $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$ represents a pair of parallel lines then condition applies

$h^2=ab \& b{g}^2=a{f}^2$

Compare the given pair of lines with  $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$ ,we get

$a=9,2h=-24,b=16,\alpha =2g,\beta =2f,c=6$

Put all given values in $b{g}^2=a{f}^2$ we get

$$\begin{gathered} 16\frac{{\alpha }^2}{4}=\frac{9{\beta }^2}{4} \\ {\left(\frac{\alpha }{\beta }\right)}^2=\frac{9}{16} \end{gathered}$$

$\frac{\alpha }{\beta }=\frac{\pm 3}{4}$.