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 $b = -a$.