Point $\mathrm{Q}$ can be obtained by using parametric form of a straight line, $\Rightarrow \mathrm{Q}\equiv \left(\sqrt{3}-\frac{\sqrt{3}r}{2}, 2-\frac{r}{2}\right)$ 

Also, point $\mathrm{Q}$ is on $\sqrt{3}x-4y+8$  so it will satisfy this equation, hence, 
$\sqrt{3}\left(\sqrt{3}-\frac{\sqrt{3}r}{2}\right)-\mathrm{4 }\left(2-\frac{r}{2}\right)+8=0$
$3-\frac{3}{2}r-8+2r+8=0$
$\Rightarrow r\left(2-\frac{3}{2}\right)=-3$

$\Rightarrow r=-6$

Hence, the length of $\mathrm{PQ}=6$