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

x^{2} + \sqrt{3} x y - y^{2} = 0

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x^{2} - \sqrt{3} x y - y^{2} = 0

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

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3 x^{2} - y^{2} = 0

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Solution

The correct option is C $\sqrt{3} x^{2} - 4 x y + \sqrt{3} y^{2} = 0$
The lines trisecting the first quadrant have slopes of $\sqrt{3}, \frac{1}{\sqrt{3}}$
The equation of line passing through origin and having slope of $\sqrt{3}$ is $(y - 0) = \sqrt{3} (x - 0)$
$\Rightarrow \sqrt{3} x - y = 0$
The equation of line passing through origin and having slope of $\frac{1}{\sqrt{3}}$ is $(y - o) = \frac{1}{\sqrt{3}} (x - 0)$
$\Rightarrow \sqrt{3} y - x = 0$
The joint equation becomes $(\sqrt{3} x - y) (\sqrt{3} y - x) = 0$
$\Rightarrow \sqrt{3} x^{2} - 4 x y + \sqrt{3} y^{2} = 0$

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