Orthogonal Trajectories/Examples/Circles Tangent to Y Axis/Proof 2

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

Expressing $(1)$ in polar coordinates, we have:
 * $(2): \quad r = 2 c \cos \theta$

Differentiating $(1)$ $\theta$ gives:
 * $(3): \quad \dfrac {\d r} {\d \theta} = -2 c \sin \theta$

Eliminating $c$ from $(2)$ and $(3)$:
 * $r \dfrac {\d \theta} {\d r} = -\dfrac {\cos \theta} {\sin \theta}$

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

Using the technique of Separation of Variables:
 * $\displaystyle \int \frac {\d r} r = \int \dfrac {\cos \theta} {\sin \theta} \rd \theta$

which by Primitive of Reciprocal and various others gives:
 * $\ln r = \map \ln {\sin \theta} + \ln 2 c$

or:
 * $r = 2 c \sin \theta$

This can be expressed in Cartesian coordinates as:
 * $x^2 + y^2 = 2 c y$

Hence the result.