Find the orthogonal trajectory to the family of curves $x^2+y^2=4c^2$ where $c$ is a constant.

Question

Differentiate w.r.t $x$:

$$x+y\frac{dx}{dy}=4c^2$$

Answer 1

$$x+yy'=0$$ turns to

$$x-\frac y{y'}=0$$ which integrates as $$y=cx.$$

This was expected as the given family is made of circles centered at the origin.