# Condition for 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 quadratic equation $E$ in two variables:

- $a x^2 + b y^2 + 2 h x y + 2 g x + 2 f y + c = 0$

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

Then:

- $a + b = 0$

That is, $E$ is of the form:

- $a \paren {x^2 - y^2} + 2 h x y + 2 g x + 2 f y + c = 0$

## Proof

From Homogeneous Quadratic Equation for Straight Lines Parallel to those Passing through Origin, $\LL_1$ and $\LL_2$ are parallel respectively to the $2$ straight lines through the origin $\LL'_1$ and $\LL'_2$ represented by the homogeneous quadratic equation:

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

As $\LL_1$ and $\LL_2$ are parallel respectively to $\LL'_1$ and $\LL'_2$, it follows that $\LL'_1$ and $\LL'_2$ are themselves perpendicular.

Hence from Condition for Homogeneous Quadratic Equation to describe Perpendicular Straight Lines, $\LL'_1$ and $\LL'_2$ are represented by the homogeneous quadratic equation:

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

That is:

- $a + b = 0$

The result follows.

$\blacksquare$

## Sources

- 1933: D.M.Y. Sommerville:
*Analytical Conics*(3rd ed.) ... (previous) ... (next): Chapter $\text {II}$. The Straight Line: $17$.