  We derive a distribution function for the position of a tagged active
particle in a slowly varying in space external potential, in a system of
interacting active particles. The tagged particle distribution has the form of
the Boltzmann distribution but with an effective temperature that replaces the
temperature of the heat bath. We show that the effective temperature that
enters the tagged particle distribution is the same as the effective
temperature defined through the Einstein relation, i.e. it is equal to the
ratio of the self-diffusion and tagged particle mobility coefficients. This
shows that this effective temperature, which is defined through a
fluctuation-dissipation ratio, is relevant beyond the linear response regime.
We verify our theoretical findings through computer simulations. Our theory
fails when an additional large length scale appears in our active system. This
length scale is associated with long-wavelength density fluctuations that
emerge upon approaching motility-induced phase separation.
