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The joint equation of the lines passing through the origin and trisecting the first quadrant is
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$\sqrt{3} x^2-4 x y+\sqrt{3} y^2=0$
$\begin{aligned} & y=\tan 30^{\circ} x \text { and } y=\tan 60^{\circ} x \\ & \Rightarrow y=\frac{1}{\sqrt{3}} x \text { and } y=\sqrt{3} \cdot x \\ & \Rightarrow x-\sqrt{3} y=0 \text { and } \sqrt{3} x-y=0\end{aligned}$
Joint equation $(x-\sqrt{3} y)(\sqrt{3} x-y)=0$ $\Rightarrow \sqrt{3} x^2-4 x y+\sqrt{3} y^2=0$
Joint equation $(x-\sqrt{3} y)(\sqrt{3} x-y)=0$ $\Rightarrow \sqrt{3} x^2-4 x y+\sqrt{3} y^2=0$
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