There's a fairly large research literature on methods for the "trust region subproblem (TRS)" of minimizing a quadratic $x^{T}Ax+b^{T}x$ subject to the constraint that $x^{T}x \leq \Delta$. This has important applications in trust region methods for optimization. There isn't any simple solution and getting a numerically stable algorithm for the precise solution of the TRS has turned out to be surprisingly hard. In most situations of interest, the optimal solution to the TRS occurs at a point where the inequality constraint is active. This is exactly your problem. Thus you'd do well to look at methods for solving the TRS.

Your problem has been studied extensively in the context of trust region methods for optimization, and there are a number of algorithms that have been developed.

See for example:

W. W. Hager, Minimizing a quadratic over a sphere. SIAM Journal on Optimization, 12:188-208, 2001.

Hager's paper gives a lemma that characterizes the solutions to your problem and a solution in terms of the eigenvalues and eigenvectors of A as well as algorithms for solving the problem. Since then there have been several other papers written on this topic, with particular attention to algorithms for solving instances where A is large and sparse- this isn't a particular issue for you.