Angle Between Two Straight Lines described by Quadratic Equation

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:

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

Then the angle $\psi$ between $\LL_1$ and $\LL_2$ is given by:

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

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:

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

From Angle Between Two Straight Lines described by Homogeneous Quadratic Equation, the angle $\psi$ between $\LL'_1$ and $\LL'_2$ is given by:

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

As:

$\LL'_1$ is parallel to $\LL_1$

$\LL'_2$ is parallel to $\LL_2$

it follows that the angle $\psi$ between $\LL_1$ and $\LL_2$ is given by:

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

$\blacksquare$

Sources

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