I was reading a paper in which they have an algebraic surface \begin{equation} F(x, y, u) = u^2(1+xy)= (x+y).(x+y-4xy+x^2y+xy^2) \end{equation} After homogenising the surface they get; \begin{equation} F(x, y, u,z) = u^2(z^2+xy)-(x+y).((x+y)z^2-4xyz+x^2y+xy^2) \end{equation} They are saying one of the singularities of this projective hypersurface is $[x:y:u:z] = [1:1:0:1]$ and this singularity is ADE type $A_1$.
I have found on the internet so far that to prove it $A_1$ type; I have to show that the singular point is analytic isomorphic to \begin{equation} x^2+y^2+z^2 =0 \end{equation} Can someone please explain how I can show that? I would be a great help if you explain the steps for the calculation in simpler words as I don't have much mathematical background knowledge in this area.