Equation of Hyperbola in Reduced Form

Theorem

Let $K$ be a hyperbola such that:

the transverse axis of $K$ has length $2 a$

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

Cartesian Frame

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

$K$ can be expressed by the equation:

$\dfrac {x^2} {a^2} - \dfrac {y^2} {b^2} = 1$

Parametric Form 1

The right-hand branch of $K$ can be expressed in parametric form as:

$\begin {cases} x = a \cosh \theta \\ y = b \sinh \theta \end {cases}$

Parametric Form 2

$K$ can be expressed in parametric form as:

$\begin {cases} x = a \sec \theta \\ y = b \tan \theta \end {cases}$

Polar Frame

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

$K$ can be expressed by the equation:

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

Also see

• Equation of Ellipse in Reduced Form