  We study the induced dynamics of an inertial tracer particle elastically
coupled to passive or active Brownian particles. We integrate out the
environment degrees of freedom to obtain generalized Langevin equation for the
tracer dynamics in both cases. In particular, we find the exact form of the
dissipation kernel and effective noise experienced by the tracer and compare it
with the phenomenological modeling of active baths used in previous studies. We
show that the second fluctuation-dissipation relation (FDR) does not hold at
early times for both cases. However, at finite times, the tracer dynamics
violate (obeys) the FDR for the active (passive) environment. We calculate the
linear response formulas in this regime for both cases and show that the
passive medium satisfies an equilibrium fluctuation response relation (FRR),
while the active medium does not -- we quantify the extent of this violation
explicitly. We show that though the active medium generally renders a
nonequilibrium description of the tracer, an effective equilibrium picture
emerges asymptotically in the small activity limit of the medium. We also
calculate the mean squared velocity and mean squared displacement of the tracer
and report how they vary with time.
