The solution i have envisioned is that I need to find the point of tangency and the vector that defined as the crossproduct of $R_x$ $R_y$ to be able to solve for the tangent plane where $u=u_0$ and $v=v_0$.
after obtaining a tangent plane equation in terms of $u_0$ and $v_0$, i should make $x=0$ and $z=0$ to be able to get the points that would be parallel to the y axis.
I tried this and got stuck with too many terms wherein i could only assume i made a wrong strategy regarding the solution of this problem.
What should be my next (first) step
*Note R is a Vector
It is asked for the plane to be parallel to the $y$ axis. This means the plane contains some line with direction vector $\mathbf{j}$ which means the normal is perpendicular to $\mathbf{j}$, i.e. its $y$ component should be zero. From what you said, I think you were trying to choose the normal to be parallel to $\mathbf{j}$ which is not correct.
In this case the equation for the $y$ component of the normal being zero is very simple, at least once you divide it through by $e^{u^2+v^2}$.