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(B) The lines trisect the $90^{\circ}$ angle in the first quadrant. Thus,the angles made by these lines with the positive $x$-axis are $30^{\circ}$ an

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$\sqrt{3}(x^{2}+y^{2})-4xy=0$

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(B) The lines trisect the $90^{\circ}$ angle in the first quadrant. Thus,the angles made by these lines with the positive $x$-axis are $30^{\circ}$ and $60^{\circ}$.Line $L_{1}$ makes an angle of $30^{\circ}$ with the $x$-axis: $y = 	an(30^{\circ})x$ $\Rightarrow y = \frac{1}{\sqrt{3}}x$ $\Rightarrow x - \sqrt{3}y = 0$.Line $L_{2}$ makes an angle of $60^{\circ}$ with the $x$-axis: $y = 	an(60^{\circ})x$ $\Rightarrow y = \sqrt{3}x$ $\Rightarrow \sqrt{3}x - y = 0$.The joint equation of these two lines is $(x - \sqrt{3}y)(\sqrt{3}x - y) = 0$.Expanding this: $\sqrt{3}x^{2} - xy - 3xy + \sqrt{3}y^{2} = 0$.$\sqrt{3}(x^{2} + y^{2}) - 4xy = 0$.

The joint equation of the lines through the origin trisecting the angles in the first and third quadrants is

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