Solving eigenvalue problem with a linear constraint

Question

I want to solve an eigenvalue problem with a linear constraint: \begin{align} \max \ & x^T A x &\\ \text{s.t.} \ & x^T x = 1& \\ & c^T x = b&\\ \end{align}

If $b = 0$, the solution can be found in Golub 1973.

How can I solve the above problem when $b \neq 0$?