Do the two equations agree? Look at the equation 
$$\tag{1}\nabla g(a)=\lambda \nabla h(a),$$ 
must be satisfied at any random point $a$ when $g$'s is on the level set $\{x: h(x)=b\}$ and has an extreme. But in the case when $g$ has a local extreme at $a$, then 
$$\tag{2}\nabla g(a)= 0.$$
Do the two equations agree with each other?

Yes, that could happen. In that case $\lambda = 0$. 
