Let P ( -1, 0), Q(0, 0) and $R\left(3,3\sqrt{3}\right)$ be three points. The equation of the bisector of the angle PQR is

$\begin{array}{l}P\left(-1,0\right)Q\left(0,0\right)R\left(3,3\sqrt{3}\right)\\ LetQSbethebi\sec torof\overline{)PQR}\end{array}$

S divides RP in the ratio QR : QP

$\begin{array}{l}\therefore QR=\sqrt{3^2+{\left(3\sqrt{3}\right)}^2}=\sqrt{9+27}=6\\ QP=\sqrt{{\left(-1\right)}^2+0^2}=1\\ \therefore QR:QP=6:1=m:n\\ R\left(3,3\sqrt{3}\right)=\left(x_1,y_1\right)P\left(-1,0\right)=\left(x_2,y_2\right)\end{array}$ 

 $\begin{array}{l}S=\left(\frac{6\left(-1\right)+1\left(3\right)}{6+1},\frac{6\left(0\right)+1\left(3\sqrt{3}\right)}{6+1}\right)\Rightarrow \left(\frac{-3}{7},\frac{3\sqrt{3}}{7}\right)\\ EquationofQSis\\ y-0=\frac{3\sqrt{3}/7}{-3/7}\left(x-0\right)\\ \Rightarrow y=-\sqrt{3}x\Rightarrow \sqrt{3}x+y=0\end{array}$