# Orthogonal Trajectories/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:

$x^2 - y^2 = c$

## 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.

$\blacksquare$