Equation of Hyperbola in Reduced Form/Polar Frame

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Let $K$ be a hyperbola such that:

the transverse axis of $K$ has length $2 a$
the conjugate axis of $K$ has length $2 b$.

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

$K$ can be expressed by the equation:

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


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:

\(\ds x\) \(=\) \(\ds r \cos \theta\)
\(\ds y\) \(=\) \(\ds r \sin \theta\)


\(\ds \dfrac {\paren {r \cos \theta}^2} {a^2} - \dfrac {\paren {r \sin \theta}^2} {b^2}\) \(=\) \(\ds 1\)
\(\ds \leadsto \ \ \) \(\ds \dfrac {r^2 \cos^2 \theta} {a^2} - \dfrac {r^2 \sin^2 \theta} {b^2}\) \(=\) \(\ds 1\)
\(\ds \leadsto \ \ \) \(\ds \dfrac {\cos^2 \theta} {a^2} - \dfrac {\sin^2 \theta} {b^2}\) \(=\) \(\ds \dfrac 1 {r^2}\)