A question on polynomials.

Question

Let a polynomial $f\in\mathbb{R}[x,y]$, and

$f(x,y)=(x^2+y^2)p(x,y)^2-q(x,y)^2$ and $p,q$ are coprime to each other.

When do, $f$ and $\frac{{\partial f}}{{\partial x}}$ and $\frac{{\partial f}}{{\partial y}}, $share a non-trivial common factor?