The joint equation of the lines passing through the origin and trisecting the first quadrant is

1. $\sqrt{3} x^2-4 x y+\sqrt{3} y^2=0$
2. $x^2-\sqrt{3} x y-y^2=0$
3. $3 x^2-y^2=0$
4. $x^2+\sqrt{3} x y-y^2=0$

Solution

$\begin{aligned} & y=\tan 30^{\circ} x \text { and } y=\tan 60^{\circ} x \\ & \Rightarrow y=\frac{1}{\sqrt{3}} x \text { and } y=\sqrt{3} \cdot x \\ & \Rightarrow x-\sqrt{3} y=0 \text { and } \sqrt{3} x-y=0\end{aligned}$ Joint equation $(x-\sqrt{3} y)(\sqrt{3} x-y)=0$ $\Rightarrow \sqrt{3} x^2-4 x y+\sqrt{3} y^2=0$

Asked in: MHT CET 2022 (08 Aug Shift 1)

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