Lagrange multipliers with $\lambda = 0$ problem We want to maximize the function $f(x_1,x_2,x_3,x_4) = \sum_{i=1}^{4}a_i^2x_i$ over the compact set $\Omega = \{x \in \mathbb R^4: |x| = 1, \langle x,a\rangle = 0\}$ where $a = (a_1,a_2,a_3,a_4)$ is some non zero vector
This is a continuous function over a compact set so it admits minimum and maximum. Thus we need to solve the system $\begin{cases}a_i^2 = 2\lambda_1 x_i +\lambda_2 a_i \\ |x|-1 = 0\\\langle x,a \rangle = 0\end{cases}$
I solved this for the case that $\lambda_1 \neq 0$. But when $\lambda_1 = 0$ I have too many variables and no way to isolate $x$ - how do I solve this?

Hint If $\lambda_1 = 0$ then $a_i^2 = \lambda_2 a_i$ which implies $\lambda_2 = a_i \Rightarrow a = (\lambda_2, \lambda_2, \lambda_2, \lambda_2)$, or $\lambda_2 = 0$, and you should be able to compute a solution.
