What to do if in an optimization problem with 2 variables and 1 equality constraint the determinant of bordered hessian is equal to zero?

Question

I have to minimize $4x+y$ subject to $\sqrt{xy}=200$, and I got a critical point $(100,400)$ with $\lambda = -4$. It seems that it is the only critical point, but the determinant of bordered hessian at this point is zero and I’m confused now.