The problem of finding the optimum current under different constraints in electrical impedance tomography is cast into a non-linear optimization problem. Optimum currents are investigated for a two-dimensional cylindrical body with a concentric or an eccentric inhomogeneity under the constraints of constant dissipated power and constant total injected current. For a concentric inhomogeneity, it is shown that the opposite drive results in a better distinguishability than the cosine current pattern under the constant-injected-current constraint. The results for the concentric case are extended to the eccentric case directly using the properties of the conformal transformation and of the constraints involved. Distinguishability and the minimum detectable object size achieved by the optimized currents are compared with the ones achieved by the cosine current pattern for conductivity distributions with the concentric and eccentric inhomogeneity.