Neumann data must integrate to zero

Let B(0,2) = {$(x,y)\in {\mathbb{R}}^2:x^2+y^2<4$}, and $\partial B$ denote the boundary of B(0,2). Assume $(\alpha ,\beta )\ne (0,0),k\in \mathbb{R}$, and u is any solution to $-\Delta u=0$ in $B(0,2),\alpha u(x,y)+\beta (\partial u/\partial \nu )(x,y)=1+(x^2+y^2)k$ on $\partial B$, where $\nu (x,y)$ is the unit outward normal to B(0,2) at $(x,y)\in \partial B$. Consider the following statements: $S_1$: If $\beta =0$, then there exists $a(x_0,y_0)\in B(0,2)$ such that |$u(x_0,y_0)$| = |1 + 4k|/|$\alpha$|. $S_2$: If $\alpha =0$, then k = −1/4. Then