Pair of Straight Lines

Represents a pair of straight lines through the origin. Every second-degree homogeneous equation (no constant term, no linear terms) factors into two linear factors through the origin, real or imaginary.

The two lines y = m₁x and y = m₂x satisfying ax² + 2hxy + by² = 0. Derived by dividing through by x² to get a quadratic in y/x = m. Valid when b ≠ 0.

The discriminant of the quadratic in m (from ax² + 2hxy + by² = 0) is 4(h²−ab). The three cases correspond to positive, zero, and negative discriminant.

Acute angle θ between the two lines of ax² + 2hxy + by² = 0 (or the general pair). Uses the identity tan θ = (m₁−m₂)/(1+m₁m₂) with m₁−m₂ expressed via the discriminant.

The two lines of ax² + 2hxy + by² = 0 are perpendicular iff a + b = 0 (sum of coefficients of x² and y² is zero). Equivalently m₁m₂ = −1 gives a/b = −1.

Combined equation of the two angle bisectors of ax² + 2hxy + by² = 0. The bisectors are always perpendicular to each other (their combined equation satisfies the perpendicularity condition: sum of x² and y² coefficients = 1 + (−1) = 0).

To determine which of the two bisectors (from x²−y² = 0 or xy = 0 form) makes the acute angle with the original lines: check the sign of h(a−b). The result avoids solving the full bisector equations.

Represents a pair of straight lines (not necessarily through the origin) when the discriminant condition Δ = 0 is satisfied. May also represent a conic if Δ ≠ 0.

The general second-degree equation represents a pair of straight lines iff this determinant (the discriminant of the conic) vanishes. When Δ ≠ 0, the equation represents a proper conic (circle, parabola, ellipse, or hyperbola).

Point of intersection of the two lines represented by the general equation, obtained by solving ∂S/∂x = 0 and ∂S/∂y = 0 simultaneously. Valid when ab − h² ≠ 0 (lines are not parallel).

The general equation represents two parallel lines iff h² = ab (same slope condition) and af² = bg² (consistency condition). When these hold, ab − h² = 0 so the point of intersection formula breaks down — the lines do not meet.

Distance between the two parallel lines represented by ax² + 2hxy + by² + 2gx + 2fy + c = 0, when h² = ab. Equivalently written as 2√((f²−bc)/(b(a+b))) using the y-coefficient form.

Combined equation of the pair of lines joining the origin to the intersection points of conic S = ax²+2hxy+by²+2gx+2fy+c = 0 and line lx+my+n = 0. Obtained by substituting 1 = −(lx+my)/n into S. The result is a homogeneous second-degree equation — always a pair of lines through the origin.