The combined equation to a pair of straight lines passing through the origin and inclined at 45⁰, 60⁰ with x-axis is

$\begin{array}{l}Let{\theta }_1={45}^0\Rightarrow m_1=\mathrm{Tan}{\theta }_1=\mathrm{Tan}{45}^0=1\\ {\theta }_2={60}^0\Rightarrow m_2=\mathrm{Tan}{\theta }_2=\mathrm{Tan}{60}^0=\sqrt{3}\\ \therefore Equationoflinesarey=m_{1}x,y=m_{2}x\end{array}$

$\begin{array}{l}\Rightarrow y=x,y=\sqrt{3}x\\ \Rightarrow x-y=0, \sqrt{3}x-y=0\\ \therefore combainedEquationis\left(x-y\right)\left(\sqrt{3}x-y\right)=0\\ \Rightarrow \sqrt{3}{x}^2-\left(\sqrt{3}+1\right)xy+y^2=0\end{array}$