Condition for Homogeneous Quadratic Equation to describe Perpendicular Straight Lines

Theorem

Let $\LL_1$ and $\LL_2$ represent $2$ straight lines in the Cartesian plane which are represented by a homogeneous quadratic equation $E$ in two variables.

Let $\LL_1$ and $\LL_2$ be perpendicular.

Then $E$ is of the form:

$a x^2 + 2 h x y - a y^2$

Proof

From Homogeneous Quadratic Equation represents Two Straight Lines through Origin, $E$ is of the form:

$a x^2 + 2 h x y + b y^2$

From Angle Between Two Straight Lines described by Homogeneous Quadratic Equation, the angle $\psi$ between $\LL_1$ and $\LL_2$ is given by:

$\tan \psi = \dfrac {2 \sqrt {h^2 - a b} } {a + b}$

When $\psi = \dfrac \pi 2$, $\tan \psi$ is undefined.

Hence:

$a + b = 0$

and so $b = -a$.

$\blacksquare$

Sources

• 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {II}$. The Straight Line: $16$. Angle between the two straight lines represented by the homogeneous equation