If $9x^2-24xy+16y^2+\alpha x+\beta y+6=0$ represents a pair of parallel lines of $1$ unit apart and one of those lines passes through $(1, 1)$ then $\frac{\alpha }{\beta }=$

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

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

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