The homogeneous equation $ax^2$ + 2hxy + $by^2$ = 0 represents a pair of straight lines through the origin. The lines are real and distinct when $h^2$ > ab, coincident when $h^2$ = ab, and imaginary when $h^2$ < ab. The individual lines have slopes given by the roots of $bm^2$ + 2hm + a = 0 (where m = $\frac{y}{x}$), so m1 + m2 = -2h/b and m1m2 = $\frac{a}{b}$. The angle between the pair is tan(theta) = 2sqrt($h^2$ - ab)/|a + b|. The lines are perpendicular when a + b = 0 (sum of coefficients of $x^2$ and $y^2$ is zero). To factorize, find the slopes from the quadratic $bm^2$ + 2hm + a = 0 and write the lines as (y - m1x)(y - m2x) = 0. The bisectors of the angle between the pair $ax^2$ + 2hxy + $by^2$ = 0 are given by $\frac{x^2 - y^2}{(a - b)}$ = $\frac{xy}{h}$.