I'm having two constraints $g_{1}$=$x+y-z+2=0$ and $g_{2}$=$z^{2}-x^{2}-y^{2}=0$ and I want to determine the point on the intersection which is closest to the origin. The question asks us to use Lagrange multiplier over two constraints, so I need to minimize $F(x,y,z)= x^{2}+y^{2}+z^{2}$
According to $∇F=λ∇g_{1}+μ∇g_{2}$ I got the following equations
$$2x=λ+μ(-2x)$$
$$2y=λ+μ(-2y)$$
$$2z=-λ+μ(2z)$$
$g_{1}$ and $g_{2}$
from the first and second equation I got $2x+2μx=2y+2μy$, which is
$x(1+μ)=y(1+μ)$
In this case do I assume $1+μ$ is not equal to $0$? In this case I can get $x=y$ and the equation is quite simple to solve.
Or do I need to consider the situation when $1+μ=0$?
There is another case for this equation to be true, that is when $1+μ=1$, that is when $μ=0$
So I need to consider all three possible solutions of $μ$?