# Equation of Hyperbola in Reduced Form

## Theorem

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

Let:

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

### Cartesian Frame

The equation of $K$ is:

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

### Parametric Form

The equation of $K$ in parametric form is:

$x = a \cosh \theta, y = b \sinh \theta$