Equation of the Bisectors of the Angle Between the Lines

The angle bisectors of the pair $ax^2+2hxy+by^2=0$ (i.e. of $y-m_1x=0$ and $y-m_2x=0$, with $m_1+m_2=-2h/b,\ m_1m_2=a/b$) are found as the locus of points equidistant from both lines. Let $P(p,q)$ be a point on a bisector; equating the perpendicular distance from $P$ to $y-m_1x=0$ with the perpendicular distance from $P$ to $y-m_2x=0$ gives

Expanding and simplifying (using $m_1+m_2=-2h/b$ and $m_1m_2=a/b$ to eliminate $m_1,m_2$ individually) reduces this to

Replacing $(p,q)$ by $(x,y)$, the combined equation of the two angle bisectors is

$\frac{x^2-y^2}{a-b}=\frac{xy}{h}.$ …