Cartesian equation of a rectangular hyperbola
Categories: coordinate systems pure mathematics
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 $a = c^2$.
See also
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