Equation of Hyperbola in Reduced Form

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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:

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


Also see