The optical design for the proposed Euro50 extremely large telescope with integrated adaptive optics (AO) is presented. For atmospheric turbulence correction, we propose using single and dual-conjugate AO systems working with natural and laser guide stars. The corrective shape of the deformable mirrors (DMs) is derived from an analytical algorithm based on minimization of the sum of the residual power spectra of the phase fluctuations seen by guide stars after correction. Predictions for performance of the Euro50 ELT with Dual-conjugate AO are given for the K band using a seven layer atmospheric model for the atmosphere at the Observatorio del Roque de los Muchachos (ORM) on La Palma. The average Strehl ratio is used to quantify the system performance for different values of actuator pitch and DM conjugation altitudes. The influence of the outer scale and telescope pointing on the RMS stroke of the DMs is presented. It is concluded that construction of such a system is feasible and that there is a need for development of a simulation tool to verify the analytical calculations. Precise knowledge of the outer scale of the atmosphere at the ORM is needed to establish the dynamical range of the mirrors.