# 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:

- $x^2 - y^2 = c$

## Proof

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

- $x \dfrac {\d y} {\d x} + y = 0$

\(\displaystyle x \frac {\d y} {\d x} + y\) | \(=\) | \(\displaystyle 0\) | |||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \frac {\d y} {\d x}\) | \(=\) | \(\displaystyle -\frac y x\) |

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

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

So:

\(\displaystyle \frac {\d y} {\d x}\) | \(=\) | \(\displaystyle \frac x y\) | |||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \int x \rd x\) | \(=\) | \(\displaystyle \int y \rd y\) | Separation of Variables | |||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle x^2\) | \(=\) | \(\displaystyle y^2 + c\) | ||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle x^2 - y^2\) | \(=\) | \(\displaystyle c\) |

Hence the result.

$\blacksquare$

## Sources

- 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): $\S 1.3$: Families of Curves. Orthogonal Trajectories: Problem $1 \ \text a$