Orthogonal Trajectories/Rectangular Hyperbolas

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Theorem

Consider the one-parameter family of curves of rectangular hyperbolas:

$(1): \quad x y = c$


Its family of orthogonal trajectories is given by the equation:

$x^2 - y^2 = c$


RectanguleHyperbolaeOrthogonalTrajectories.png


Proof

Differentiating $(1)$ with respect to $x$ gives:

$x \dfrac {\mathrm d y} {\mathrm d x} + y = 0$
\(\displaystyle x \frac {\mathrm d y} {\mathrm d x} + y\) \(=\) \(\displaystyle 0\)
\(\displaystyle \implies \ \ \) \(\displaystyle \frac {\mathrm d y} {\mathrm d x}\) \(=\) \(\displaystyle - \frac y x\)


Thus from Orthogonal Trajectories of One-Parameter Family of Curves, the family of orthogonal trajectories is given by:

$\dfrac {\mathrm d y} {\mathrm d x} = \dfrac x y$


So:

\(\displaystyle \frac {\mathrm d y} {\mathrm d x}\) \(=\) \(\displaystyle \frac x y\)
\(\displaystyle \implies \ \ \) \(\displaystyle \int x \, \mathrm d x\) \(=\) \(\displaystyle \int y \, \mathrm d y\) Separation of Variables
\(\displaystyle \implies \ \ \) \(\displaystyle x^2\) \(=\) \(\displaystyle y^2 + c\)
\(\displaystyle \implies \ \ \) \(\displaystyle x^2 - y^2\) \(=\) \(\displaystyle c\)

Hence the result.

$\blacksquare$


Sources