Orthogonal Trajectories/Examples/Circles Tangent to Y Axis/Proof 1

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

from which:
 * $\dfrac {\mathrm d y} {\mathrm d x} = \dfrac {y^2 - x^2} {2 x y}$

Thus from Orthogonal Trajectories of One-Parameter Family of Curves, the family of orthogonal trajectories is given by:
 * $\dfrac {\mathrm d y} {\mathrm d x} = \dfrac {2 x y} {x^2 - y^2}$