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 = y/x), so m1 + m2 = -2h/b and m1m2 = 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 (x^2 - y^2)/(a - b) = xy/h.