In this paper we give a complete description of how the Jacobi theory of conjugate points can be extended to a regular linear-quadratic control problem where the state end-points are jointly constrained to belong to a subspace of R-n x R-n and there is a linear pointwise state-control constraint, We introduce also the definition of semiconjugate point which describes a distinctive feature of these problems and state the corresponding necessary and sufficient conditions for the quadratic form to be nonnegative or coercive. In the case in which the constraints and the costs act separately on the initial and final points we give equivalent characterizations of the coercivity of the quadratic form by means of the solutions of an associated Riccati equation in both the controllable and uncontrollable case.