Equation of Rectangular Hyperbola in Standard Form/Parametric Form

Theorem

Let $\KK$ be a rectangular hyperbola in standard form.

$\KK$ can be expressed in parametric form as:

$\begin {cases} x = c t \\ y = \dfrac c t \end {cases}$

Proof

Let the point $\tuple {x, y}$ satisfy the equations:

\(\ds x\)

\(=\)

\(\ds c t\)

\(\ds y\)

\(=\)

\(\ds \dfrac c t\)

Then:

\(\ds x y\)

\(=\)

\(\ds c t \times \dfrac c t\)

\(\ds \)

\(=\)

\(\ds c^2\)

$\blacksquare$

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): hyperbola
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): hyperbola