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