Optimal deployment of unmanned aerial vehicles (UAVs) as communication relays is studied for fixed-rate variable-power systems. The considered setup is a set of ground transmitters (GTs) wishing to communicate with a set of ground receivers (GRs) through the UAVs. Each GT-GR pair communicates through only one selected UAV and have no direct link. Two different UAV selection scenarios are studied: In centralized selection, a decision center assigns an optimal UAV depending on the locations of all terminals. In distributed selection, a GT selects its relaying UAV using only the local knowledge of its distances to the UAVs. For both selection scenarios, the optimal tradeoff between the UAV and GT power consumptions are determined using tools from quantization theory. Specifically, the two extremal regimes of one UAV and very large number of UAVs are analyzed for a path loss exponent of $2$. Numerical optimization of UAV locations are also discussed. Simulations are provided to confirm the analytical findings.