Equation of Hyperbola in Reduced Form/Cartesian Frame/Parametric Form 1

Theorem

Let $K$ be a hyperbola such that:

the transverse axis of $K$ has length $2 a$

the conjugate axis of $K$ has length $2 b$.

Let $K$ be aligned in a cartesian plane in reduced form.

The right-hand branch of $K$ can be expressed in parametric form as:

$\begin {cases} x = a \cosh \theta \\ y = b \sinh \theta \end {cases}$

Proof

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

\(\ds x\)

\(=\)

\(\ds a \cosh \theta\)

\(\ds y\)

\(=\)

\(\ds b \sinh \theta\)

Then:

\(\ds \frac {x^2} {a^2} - \frac {y^2} {b^2}\)

\(=\)

\(\ds \frac {\paren {a \cosh \theta}^2} {a^2} - \frac {\paren {b \sinh \theta}^2} {b^2}\)

\(\ds \)

\(=\)

\(\ds \frac {a^2} {a^2} \cosh^2 \theta - \frac {b^2} {b^2} \sinh^2 \theta\)

\(\ds \)

\(=\)

\(\ds \cosh^2 \theta - \sinh^2 \theta\)

\(\ds \)

\(=\)

\(\ds 1\)

Difference of Squares of Hyperbolic Cosine and Sine

$\blacksquare$

Sources

• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): hyperbolic function

• Weisstein, Eric W. "Hyperbola." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Hyperbola.html