Cartesian equation of a rectangular hyperbola

Martin McBride

We can convert the parametric equation of a hyperbola into a Cartesian equation (one involving only $x$ and $y$ but not $t$). Here are the parametric equations:

$$ \begin{align} x = c t\newline y = \frac{c}{t} \end{align} $$

We can eliminate $t$ from these equations simply by multiplying $x$ and $y$:

$$ \begin{align} x y &= c t \times \frac{c}{t}\newline x y &= \frac{c^2 t}{t}\newline x y &= c^2 \end{align} $$

This can also be written as:

$$ y = \frac{c^2}{x} $$

A rectangular hyperbola is a reciprocal curve

The Cartesian form of the hyperbola is a reciprocal curve of the form:

$$ y = \frac{a}{x} $$

where $a = c^2$.