Orthogonal Trajectories/Examples/Rectangular Hyperbolas

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
We use the technique of formation of ordinary differential equation by elimination.

Differentiating $(1)$ $x$ gives:
 * $x \dfrac {\d y} {\d x} + y = 0$

Thus from Orthogonal Trajectories of One-Parameter Family of Curves, the family of orthogonal trajectories is given by:
 * $\dfrac {\d y} {\d x} = \dfrac x y$

So:

Hence the result.