Asymptotes to Hyperbola in Reduced Form

Theorem
Let $\KK$ be a hyperbola embedded in a cartesian plane in reduced form with the equation:
 * $\dfrac {x^2} {a^2} - \dfrac {y^2} {b^2} = 1$

$\KK$ has two asymptotes which can be described by the equation:
 * $\dfrac {x^2} {a^2} - \dfrac {y^2} {b^2} = 0$

that is:
 * $y = \pm \dfrac b a x$

Proof
From Equation of Hyperbola in Reduced Form: Polar Frame, $\KK$ can be described in polar coordinates as:


 * $\dfrac {\cos^2 \theta} {a^2} - \dfrac {\sin^2 \theta} {b^2} = \dfrac 1 {r^2}$

When $\theta = 0$ we have that $r = a$.

As $\theta$ increases, $\cos^2 \theta$ decreases and $\sin^2 \theta$ increases.

Hence $\dfrac 1 {r^2}$ decreases and $r$ increases as a consequence.

This continues until:
 * $\dfrac {\cos^2 \theta} {a^2} - \dfrac {\sin^2 \theta} {b^2} = 0$

that is:

This can be expressed in the form:
 * $b^2 x^2 - a^2 y^2 = 0$

This is a homogeneous quadratic equation in $2$ variables of the form:
 * $a' x^2 + 2 h' x y + b' y^2 0 $

where:
 * $h' = 0$
 * $a' = b^2$
 * $b' = a^2$

From Homogeneous Quadratic Equation represents Two Straight Lines through Origin, this is the equation for $2$ straight lines through the origin:


 * $y = \pm \dfrac b a x$