We present a method for studying amplification of electromagnetic modes in active, circularly symmetric waveguides with hyperbolic dispersion. Using this method, we obtain a closed-form expression for the modal threshold condition. We find that modal amplification is possible in a region of the radius-wavelength phase-space with small enough radius so that propagation of the mode is permitted while modal energy and phase counter-propagate. At telecommunication frequencies, such a situation is achievable only when the absolute value of the real metal permittivity exceeds that of the active dielectric. We validate our theoretical conclusions with numerical simulations that explain the threshold condition in terms of an energy balance between the longitudinal and radial components of the electric field.