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

Also, point $\mathrm{Q}$ is on $\sqrt{3}\text{x}-4\text{y}+8$  so it will satisfy this equation, hence, 

$\sqrt{3}\left(\sqrt{3}-\frac{\sqrt{3}\text{r}}{2}\right)-\mathrm{4 }\left(2-\frac{\text{r}}{2}\right)+8=0$

$3-\frac{3}{2}\text{r}-8+2\text{r}+8=0$

$$\begin{gathered} \text{⇒r}\left(2-\frac{3}{2}\right)=-3 \\ \Rightarrow \mathrm{r}=-6 \end{gathered}$$

Hence, the length of $\mathrm{PQ}=6$