Question

Find the separate equations of the lines represented by the following equation:-
$x^2+2xy\tan \alpha -y^2=0$

Answer 1

$x^2+2xy\tan \alpha -y^2=0$

Dividing by $y^2$

$(\frac{x}{y}{)}^2+2(\frac{x}{y})\tan \alpha -1=0$

$\therefore \frac{x}{y}=\frac{-2\tan \alpha \pm \sqrt{4{\tan }^2\alpha -4\times 1\times 1}}{2\times 1}$

$=\frac{-2\tan \alpha \pm 2\sqrt{{\tan }^2\alpha -1}}{2}$

$=-\tan \alpha \pm \sec \alpha$

$(\frac{x}{y}=(\sec \alpha -\tan \alpha )$ and $\left(\frac{x}{y}\right)=-(\tan \alpha +\sec \alpha )$

The separate equations of the lines are

$(\sec \alpha -\tan \alpha )x+y=0$ and $(\sec \alpha +\tan \alpha )x-y=0$