The joint equation of pair of lines through the origin, each of which makes an angle of $30^{\circ}$ with Y…
The joint equation of pair of lines through the origin, each of which makes an angle of $30^{\circ}$ with Y -axis, is
$3 x^2-y^2=0$
$x^2-3 y^2=0$
$3 x^2+y^2=0$
$x^2+3 y^2=0$
Solution
Let OA and OB be the required lines.
$\therefore \quad$ angles made by OA and OB with X -axis are $60^{\circ}$ and $120^{\circ}$ respectively.
$\therefore \quad$ Their equations are $y=\sqrt{3} x$ and $y=-\sqrt{3} x$ i.e., $\sqrt{3} x-y=0$ and $\sqrt{3} x+y=0$
$\therefore \quad$ The joint equation of the lines is
$(\sqrt{3} x-y)(\sqrt{3} x+y)=0 \Rightarrow 3 x^2-y^2=0$
