Combined equation of pair of lines passing through origin and inclined at 30⁰ and 60⁰ respectively with x-axis is

$\begin{array}{l}Let{\theta }_1={30}^{0}and{\theta }_2={60}^0\Rightarrow m_1=\mathrm{Tan}{30}^0\Rightarrow \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\sqrt{3}$}\right.,m_2=\mathrm{Tan}{60}^0=\sqrt{3}\\ Equationoflinespas\sin gthrough\left(0,0\right)isy=m_1{x}_1,y=m_{2}x\\ \Rightarrow y=\frac{x}{\sqrt{3}},\therefore y=\sqrt{3}x\\ \Rightarrow x-\sqrt{3}y=0,\sqrt{3}x-y=0\\ \therefore CombaindedEquationis\left(x-\sqrt{3}y\right)\left(\sqrt{3}x-y\right)=0\\ \Rightarrow \sqrt{3}{x}^2-4xy+\sqrt{3}{y}^2=0\end{array}$