  We explore the possibility of formation of $\Delta$ baryons (1232 MeV) in
neutron star matter in an effective chiral model within the relativistic
mean-field framework. With variation in delta-meson couplings, consistent with
the constraints imposed on them, the resulting equation of state is obtained
and the neutron star properties are calculated for static and spherical
configuration. Within the framework of our model the critical densities of
formation of $\Delta$s and the properties of neutron stars are found to be very
sensitive to the iso-vector coupling compared to the scalar or vector
couplings. We revisit the $\Delta$ puzzle and look for the possibility of phase
transition from non-strange hadronic matter (including nucleons and $\Delta$s)
to deconfined quark matter, based on QCD theories. The resultant hybrid star
configurations satisfy the observational constraints on mass from the most
massive pulsars PSR J1614-2230 and PSR J0348+0432 in static condition obtained
with the general hydrostatic equilibrium based on GTR. Our radius estimates are
well within the limits imposed from observational analysis of QLMBXs. The
obtained values of $R_{1.4}$ are in agreement with the recent bounds specified
from the observation of gravitational wave (GW170817)from binary neutron star
merger. The constraint on baryonic mass from study of binary system PSR
J0737-3039 is also satisfied with our hybrid equation of state.
  \noindent{Keywords: Delta baryons, Quark matter, Phase transition, Equation
of State, Neutron Stars, Hybrid Stars}
