Equation of Hyperbola in Reduced Form/Cartesian Frame/Parametric Form

Theorem
Let $K$ be aligned in a cartesian plane in reduced form.

$K$ can be expressed in parametric form as:
 * $\begin {cases} x = a \cosh \theta \\ y = b \sinh \theta \end {cases}$

Proof
Let the point $\tuple {x, y}$ satisfy the equations:
 * $x = a \cosh \theta$
 * $y = b \sinh \theta$

Then: