Families of orthogonal curves to parabola.

Question

In a typical ODE course, we learn that families of orthogonal curves to parabola $y=Ax^2$ are given by families of ellipses, given by $x^2/2+y^2=c^2$. However, there is this thing called parabolic coordinates which are families of parabolas orthogonal to each other. It seems like either family of ellipses and another family of parabola can be orthogonal to families of parabolas. Can there exist more than one families of orthogonal curves?

Answer 1

(see figure below).

No, (setting apart pathological cases) there is unicity of the family of orthogonal curves to a given family of curves depending on a parameter (in a differentiable way).

Indeed, in your case :

• the family of parabolas with equations $y=ax^2$ have the origin as a common vertex whereas,

• the family of parabolas you mention (self orthogonal) with equations $y=-\frac{1}{2k}x^2+\frac{k}{2}$ have the origin as a common focus (blue for the initial family with $k>0$, red for the orthogonal family with $k<0$).

Explanation for the second family :

The directrix has equation $y=k$ ; expressing that the generic point $M=(x,y)$ must be at the same distance from the origin and from the directrix gives condition $x^2+y^2=(y-k)^2$. Expanding it, one obtains the given equation.