Equation of Hyperbola in Reduced Form/Polar Frame

Theorem

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:

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

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:

\(\ds x\)

\(=\)

\(\ds r \cos \theta\)

\(\ds y\)

\(=\)

\(\ds r \sin \theta\)

Hence:

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

\(=\)

\(\ds 1\)

\(\ds \leadsto \ \ \)

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

\(=\)

\(\ds 1\)

\(\ds \leadsto \ \ \)

\(\ds \dfrac {\cos^2 \theta} {a^2} - \dfrac {\sin^2 \theta} {b^2}\)

\(=\)

\(\ds \dfrac 1 {r^2}\)

$\blacksquare$

Sources

• 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {V}$. The Hyperbola: $3$. Asymptotes