Find the single equation of the two lines through the origin and perpendicular to the each lines represented by $ax^2+2hxy+by^2=0$
I tried the factorization of the given equation but it was fail..
Given $ax^2 + 2hxy + by^2 = 0$ is equation of two lines passing through origin.
Let the equation of lines be
$y = m_1x$ and $y = m_2x$
Dividing the equation by
$x^2$
$$b(\frac yx)^2 + 2h(\frac yx) + a = 0$$
We know
$\frac yx$
is the slope of the lines, say $m$
Thus,
$$bm^2 + 2hm + a = 0$$
Its clearly a quadratic equation. Solving
$$m = \frac{-2h \pm \sqrt{4h^2 - 4ab}}{2b}$$
or
$$m = \frac{-h \pm \sqrt{h^2 - ab}}{b}$$
Thus equation being
$$y = \frac{-h \pm \sqrt{h^2 - ab}}{b}x$$