  We establish various extensions of the convexity Dines theorem for a
(joint-range) pair of inhomogeneous quadratic functions. If convexity fails we
describe those rays for which the sum of the joint-range and the ray is convex.
These results are suitable for dealing nonconvex inhomogeneous quadratic
optimization problems under one quadratic equality constraint. As applications
of our main results, different sufficient conditions for the validity of
S-lemma (a nonstrict version of Finsler's theorem) for inhomogenoeus quadratic
functions, is presented. In addition, a new characterization of strong duality
under Slater-type condition is established.
