Question

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

• A. $x^{2}+\sqrt{3}xy -y^{2} = 0$
• B. $x^{2}-\sqrt{3}xy -y^{2} = 0$
• C. $\sqrt{3}x^{2} -4xy +\sqrt{3}y^{2} = 0$
• D. $3x^{2} -y^{2} = 0$

Answer 1

Answer: (C)

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

Answer 2

In a trisection of lines in quadrant, angle $90^{\circ}$ is divided into three parts and each part contain $30^{\circ}$
$\therefore$ Equation of line $OB$ is

$y=tan \,30^{\circ} x$
$\Rightarrow y =\frac{1}{\sqrt{3}} x$
$x-\sqrt{3} y =0$
And equation of line $OC$ is
$y=tan \,60^{\circ} x$
$\Rightarrow y =\sqrt{3x}$
$(\sqrt{3} x-y) =0$
$\therefore$ Combined equation is
$(x-\sqrt{3} y)(\sqrt{3} x-y)=0$
$\Rightarrow \sqrt{3} x^{2}-xy-3xy+\sqrt{3} y^{2}=0$
$\Rightarrow \sqrt{3} x^{2}-4xy+\sqrt{3} y^{2}=0$