# Equation of Hyperbola in Reduced Form

Jump to navigation
Jump to search

## 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

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

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

### Polar Frame

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

$K$ can be expressed by the equation:

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