Pair of Straight Lines — Maths STD 12 Science — Question

Therefore the equation of a line (l1) passing through the origin and having slope is

$\begin{aligned} & y=(1+\sqrt{3}) x \\ & \therefore(1+\sqrt{3}) x-y=0 . .(1)\end{aligned}$

Similarly, the equation of the line (l2) passing through the origin and having slope is

$\begin{aligned} & y=(1-\sqrt{3}) x \\ & \therefore(1-\sqrt{3}) x-y=0\end{aligned}$

From (1) and (2) the required combined equation is

$[(1+\sqrt{3}) x-y][(1-\sqrt{3}) x-y]=0$

$\therefore(1+\sqrt{3}) \times[(1-\sqrt{3}) x-y]-y[(1-\sqrt{3}) x-y]=0$

$\therefore(1-\sqrt{3})(1+\sqrt{3}) x^2-(1+\sqrt{3}) x y-(1-\sqrt{3}) x y+y^2=0$

$\therefore\left((1)^2-(\sqrt{3})^2\right) x^2-[(1+\sqrt{3})+(1-\sqrt{3})] x y+y^2=0$

$\begin{aligned} & \therefore(1-3) x^2-2 x y+y^2=0 \\ & \therefore-2 x^2-2 x y+y^2=0 \\ & \therefore 2 x^2+2 x y-y^2=0\end{aligned}$

This is the required combined equation.