lagrange multiplier determinant

Question

Can someone please explain why in my textbook they write lagrange multiplier like this : $$\begin{vmatrix} \frac{\partial f}{\partial x}& \frac{\partial f}{\partial y}\\ \\ \frac{\partial g}{\partial x} &\frac{\partial g}{\partial y} \end{vmatrix}=0 $$ I don't understand where did this determinant come from and why is this determinant equal to zero? I just know that the lagrange multiplier is : $f_x(x_0,y_0)=\lambda g_x(x_0,y_0),f_y(x_0,y_0)=\lambda g_y(x_0,y_0)$

Answer 1

Note that $f_i=\lambda g_i\implies f_xg_y=\lambda g_xg_y=g_xf_y$, so the determinant is $f_xg_y-g_xf_y=0$.