Bisectors of Angles between Two Straight Lines/Homogeneous Quadratic Equation Form

Theorem

Consider the homogeneous quadratic equation:

$(1): \quad a x^2 + 2 h x y + b y^2 = 0$

representing two straight lines through the origin.

Then the homogeneous quadratic equation which represents the angle bisectors of the angles formed at their point of intersection is given by:

$h x^2 - \paren {a - b} x y - h y^2 = 0$

Proof

From Angle Bisectors are Harmonic Conjugates, the two angle bisectors are harmonic conjugates of the straight lines represented by $(1)$.

From Condition for Homogeneous Quadratic Equation to describe Perpendicular Straight Lines, these angle bisectors can be described by the homogeneous quadratic equation:

$x^2 + 2 \lambda x y - y^2 = 0$

From Condition for Pairs of Lines through Origin to be Harmonic Conjugates: Homogeneous Quadratic Equation Form:

$-a + b - 2 \lambda h = 0$

Hence:

$\lambda = -\dfrac {a + b} {2 h}$

The result follows.

$\blacksquare$

Sources

• 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {II}$. The Straight Line: $22$. Equation of the bisectors of the angles between the two straight lines $a x^2 + 2 h x y + b y^2 = 0$