## Cartesian equation of a rectangular hyperbola

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:

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

We can eliminate `$t$`

from these equations simply by multiplying `$x$`

and `$y$`

:

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

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 `$a = c^2$`

.