We study the linear optical response in a hybrid nanostructure composed of a semiconductor quantum dot and a metal shell nanoparticle. We analyze the case that the nanostructure interacts with an incident electromagnetic field with polarization parallel to the symmetry axis of the nanosystem. We derive the nonlinear density matrix equations, in the rotating wave approximation, under quasistatic response of the system and use a series expansion method to obtain analytical functions of the linear susceptibility of both components of the hybrid nanostructure. The derived susceptibilities particularly depend on the average radius and the thickness of the metal nanoshell, on the material of the dielectric core and on the distance between the semiconductor quantum dot and metal nanoshell. We can identify two distinct types of hybrid exciton states, each one emerging under different circumstances. The regime in which the first type of the hybrid exciton state arises requires low values of the dielectric constant and the radius of the dielectric core. This state is characterized by a significantly amplified gain without population inversion resulting from non-radiative energy transfer between the two nanoparticles, as well as to a quenched absorption resonance associated with a suppressed exciton lifetime. The second type of the hybrid exciton state appears when the exciton-plasmon coupling is quite weak, either due to the reduction of the thickness of the metal nanoshell, or the use of a core dielectric material with a high dielectric permittivity. These conditions favor the suppression of the gain arising on the absorption spectrum of the metal nanoshell and the creation of an amplified absorption resonance that corresponds to an enhanced exciton lifetime.