Orthogonal Trajectories/Concentric Circles

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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


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

$2 x + 2 y \dfrac {\mathrm d y} {\mathrm d x} = 0$

from which:

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

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

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


Using the technique of Separation of Variables:

$\displaystyle \int \frac {\mathrm d y} y = \int \frac {\mathrm d x} x$

which by Primitive of Reciprocal gives:

$\ln y = \ln x + \ln c$

or:

$y = c x$

Hence the result.

$\blacksquare$


Sources