I want to minimize the following function
$$f(x,y,z) = x^2+ y^2 + z^2$$
subject to the constraint
$$y^2 = 9 + xz$$
I have written a Lagrangian
$$L(x,y,z) = x^2+ y^2 + z^2 + \lambda (y^2 -xz - 9)$$
My first order conditions are $$[x]: \frac{2x}{z} = \lambda$$ $$[y]: \frac{-2y}{2y} = \lambda = -1$$ $$[z]: \frac{2z}{x}=\lambda$$
But no solutions exist for this and I've never seen a Lagrangian multiplier problem where one of the constraints cancels out all the variables and leaves the multiplier as a number (here the $y$ condition).
Why is the normal procedure not working and giving me FOCs without solutions?
You may not divide by $x,y,z$ as they can be zero. The FOC: $$\begin{cases} L_x=2x-z\lambda = 0 \\ L_y=2y+2y\lambda=0 \\ L_z=2z-x\lambda=0 \\ L_{\lambda}=y^2-xz-9=0\end{cases}.$$ From $(2)$ we get: $$2y(1+\lambda)=0 \Rightarrow y=0 \ \text{or} \ \lambda=-1.$$ Adding $(1)$ and $(3)$ we get: $$(x+z)(2-\lambda)=0 \Rightarrow x=-z \ \text{or} \ \lambda =2.$$ So, consider the three cases: $1) \ y=0,x=-z; \ 2) \ y=0,\lambda =2; \ 3) \ \lambda=-1,x=-z$.
The answers are: $1) \ (-3,0,3), (3,0,-3) \ \text{saddle}; 2) \ \emptyset; 3) \ (0,-3,0), (0,3,0) \ \text{min}. $