Equation of Hyperbola in Reduced Form/Cartesian Frame/Parametric Form 2

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.

$K$ can be expressed in parametric form as:

$\begin {cases} x = a \sec \theta \\ y = b \tan \theta \end {cases}$

Proof

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

\(\ds x\)

\(=\)

\(\ds a \sec \theta\)

\(\ds y\)

\(=\)

\(\ds b \tan \theta\)

Then:

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

\(=\)

\(\ds \frac {\paren {a \sec \theta}^2} {a^2} - \frac {\paren {b \tan \theta}^2} {b^2}\)

\(\ds \)

\(=\)

\(\ds \frac {a^2} {a^2} \sec^2 \theta - \frac {b^2} {b^2} \tan^2 \theta\)

\(\ds \)

\(=\)

\(\ds \sec^2 \theta - \tan^2 \theta\)

\(\ds \)

\(=\)

\(\ds 1\)

Difference of Squares of Secant and Tangent

$\blacksquare$

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

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