Can this maximization problem be uniquely solved?

Question

Having the equation $$a^Tx=b$$where $a,x \in \mathbb{R}^n$, $b$ is a number. Can I solve $x$ in terms of $a,b$?

Ultimate goal is to find the maximum of $$\frac{1}2 x^TPx$$ given $a^Tx=b$, where $P$ is a symmetric positive definite matrix.

Answer 1

Solve first order conditions for the Lagrangian problem, then optimal could be calculated.

Answer 2

This maximization problem is equivalent to maximizing the norm of $x$ with respect to the inner product induced by $P$ on $\mathbb{R}^n$.

On the other hand, given $a$ and $b$ the equation $a^{T}x=b$ defines an affine plane on $\mathbb{R}^n$, so generally there should be no such maximum, ie, the value of $\frac12x^TPx$ should be unbounded on that plane.