The joint equation of two lines passing throu | vedclass

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(C) The lines pass through the origin and make an angle of $30^{\circ}$ with the $Y$-axis. The angles these lines make with the positive $X$-axis are

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

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(C) The lines pass through the origin and make an angle of $30^{\circ}$ with the $Y$-axis. The angles these lines make with the positive $X$-axis are $90^{\circ}-30^{\circ}=60^{\circ}$ and $90^{\circ}+30^{\circ}=120^{\circ}$. The slopes of the lines are $m_{1}=	an 60^{\circ}=\sqrt{3}$ and $m_{2}=	an 120^{\circ}=-\sqrt{3}$. The equations of the lines are $y=\sqrt{3}x$ and $y=-\sqrt{3}x$,which can be written as $\sqrt{3}x-y=0$ and $\sqrt{3}x+y=0$. The joint equation is $(\sqrt{3}x-y)(\sqrt{3}x+y)=0$. Expanding this,we get $(\sqrt{3}x)^{2}-y^{2}=0$,which simplifies to $3x^{2}-y^{2}=0$.

The joint equation of two lines passing through the origin,each making an angle of $30^{\circ}$ with the $Y$-axis,is

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