Orthogonal Trajectories/Examples/Concentric Circles

Theorem
Consider the one-parameter family of curves:
 * $(1): \quad x^2 + y^2 = c$

Its family of orthogonal trajectories is given by the equation:
 * $y = c x$

Proof

 * ConcentricCirclesOrthogonalTrajectories.png

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

Differentiating $(1)$ $x$ gives:
 * $2 x + 2 y \dfrac {\d y} {\d x} = 0$

from which:
 * $\dfrac {\d y} {\d x} = -\dfrac x y$

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

Using the technique of Separation of Variables:
 * $\displaystyle \int \frac {\d y} y = \int \frac {\d x} x$

which by Primitive of Reciprocal gives:
 * $\ln y = \ln x + \ln c$

or:
 * $y = c x$

Hence the result.