  In this paper, we study the extended trust region subproblem (eTRS) in which
the trust region intersects the unit ball with a single linear inequality
constraint. By reformulating the Lagrangian dual of eTRS as a two-parameter
linear eigenvalue problem, we state a necessary and sufficient condition for
its strong duality in terms of an optimal solution of a linearly constrained
bivariate concave maximization problem. This results in an efficient algorithm
for solving eTRS of large size whenever the strong duality is detected.
Finally, some numerical experiments are given to show the effectiveness of the
proposed method.
