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Find the orthogonal trajectories of the family xy = C

by RoRi

Find the orthogonal trajectories of the family of curves given by

    \[ xy = C. \]

From the given family of curves, we find a differential equation the curves all satisfy,

    \[ xy = C \quad \implies \quad y' = -\frac{C}{x^2} = -\frac{y}{x}. \]

Letting f(x,y) = -\frac{y}{x}, we know the orthogonal trajectories are the curves which satisfy a differential equation

    \[ y' = -\frac{1}{f(x,y)} \quad \implies \quad y' = \frac{x}{y}. \]

Therefore, the orthogonal trajectories are the curves,

    \begin{align*} y' = \frac{x}{y} && \implies && yy' &= x \\ && \implies && \frac{1}{2}y^2 &= \frac{1}{2} x^2 + C \\ && \implies && x^2 - y^2 = C, \end{align*}

where C is an arbitrary constant.

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