Equation of Hyperbola in Reduced Form/Polar Frame

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

$K$ can be expressed by the equation:
 * $\dfrac {\cos^2 \theta} {a^2} - \dfrac {\sin^2 \theta} {b^2} = \dfrac 1 {r^2}$

Proof
Let the polar plane be aligned with its corresponding Cartesian plane in the conventional manner.

We have that
 * $\dfrac {x^2} {a^2} - \dfrac {y^2} {b^2} = 1$

From Conversion between Cartesian and Polar Coordinates in Plane:

Hence: