I think the question is asking the case where $\lambda =0$.

It is only part of the story. Consider plane $(\alpha,\beta)$. After you found when $\lambda=0$ is an e.v. you got an equation to $(\alpha,\beta)$ and it describes a hyperbola, which breaks the plane into 3 zones. Since $\lambda_n=\lambda_n(\alpha,\beta)$ depend on $(\alpha,\beta)$ continuously in each of those zones the number of negative eigenvalues is the same. This number changes when you go from one zone to another and some eigenvalue crosses $0$.

One can use hint: $\lambda_n$ monotone with respect to each of arguments. If $alpha>0,\beta>0$ then there are no negative e.v.