Angle Between Two Straight Lines described by Homogeneous Quadratic Equation

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:


 * $E: \quad a x^2 + 2 h x y + b y^2$

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
Let us rewrite $E$ as follows:


 * $b y^2 + 2 h x y + a x^2 = b \paren {y - \mu_1 x} \paren {y - \mu_2 x}$

Thus from Homogeneous Quadratic Equation represents Two Straight Lines through Origin:
 * $\LL_1$ and $\LL_2$ are represented by the equations $y = \mu_1 x$ and $y = \mu_2 x$ respectively.

From Sum of Roots of Quadratic Equation:


 * $\mu_1 + \mu_2 = -\dfrac {2 h} b$

From Product of Roots of Quadratic Equation:


 * $\mu_1 \mu_2 = \dfrac a b$

From Angle between Straight Lines in Plane:
 * $\tan \psi = \dfrac {\mu_1 - \mu_2} {1 + \mu_1 \mu_2}$

We have that: