  $\mathcal{N}$ emitters collectively coupled to a quantised cavity mode are
described by the Tavis-Cummings model. We present complete analytical solution
of the model in the presence of inhomogeneous couplings and energetic disorder.
We derive the exact expressions for the bright and the dark sectors that
decouple the disordered model and find that, in the thermodynamic limit, the
energetic disorder transforms the bright sector to Fano's model that can be
easily solved. We thoroughly explore the effects of energetic disorder assuming
a Gaussian distribution of emitter transition energies. We compare the Fano
resonances in optical absorption and inelastic electron scattering both in the
weak and the strong coupling regimes. We study the evolution of the optical
absorption with an increase in the disorder strength and find that it changes
the lower and upper polaritons to their broadened resonances that finally
transform to a single resonance at the bare cavity photon energy, thus taking
the system from the strong to the weak coupling regime. Interestingly, we learn
that the Rabi splitting can exist even in the weak coupling regime while the
polaritonic peaks in the strong coupling regime can represent almost excitonic
states at intermediate disorder strengths. We also calculate the photon Green's
function to see the effect of cavity leakage and non-radiative emitter losses
and find that the polariton linewidth exhibits a minimum as a function of
detuning when the cavity leakage is comparable to the Fano broadening.
