# Equation of Hyperbola in Reduced Form/Cartesian Frame/Parametric 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 $2 b$.

The equation of $K$ in parametric form is:

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

## Proof

Let the point $\left({x, y}\right)$ satisfy the equations:

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

Then:

 $\displaystyle \frac {x^2} {a^2} - \frac {y^2} {b^2}$ $=$ $\displaystyle \frac {\left({a \cosh \theta}\right)^2} {a^2} - \frac {\left({b \sinh \theta}\right)^2} {b^2}$ $\displaystyle$ $=$ $\displaystyle \frac {a^2} {a^2} \cosh^2 \theta - \frac {b^2} {b^2} \sinh^2 \theta$ $\displaystyle$ $=$ $\displaystyle \cosh^2 \theta - \sinh^2 \theta$ $\displaystyle$ $=$ $\displaystyle 1$ Difference of Squares of Hyperbolic Cosine and Sine

$\blacksquare$