How Does Charge Distribution Affect Electrostatic Field Dependence?

[ShaddollDa9u]

Homework Statement

"Find the direction and the variables which the electrostatic field depends on at all points of the plane (xOy) uniformly charged with the density of positive charges ϱ"

The Attempt at a Solution

So first of all, I have to study the Invariances of symmetry. I tried to switch to the cylindrical coordinate system (Ur, Uθ, Uz) and I observe that there is an Rotational symmetry following Uθ and a Translational symmetry following Uz, which made me conclude that the electrostatic field only depends on r.

2. Homework Equations

However, my teacher found that there is a Rotational symmetry following Uθ and a Translational symmetry following Ur, which means that the field only depends on z.
Shouldn't it be the opposite ?? How did he find so ??

Thank you

 

[DrClaude]

All the charges are in the xOy plane. How does that lead to a translation symmetry in z?