We investigate two closely related setups. In the first one we consider a TASEP-style system of particles with specified initial and final configurations. The probability of each history of the system is assumed to be equal. We show that the rescaled trajectory of the \emph{second class particle} converges (as the size of the system tends to infinity) to a random arc of an ellipse. In the second setup we consider a uniformly random Young tableau of square shape and look for typical (in the sense of probability) sliding paths and evacuation paths in the asymptotic setting as the size of the square tends to infinity. We show that the probability distribution of such paths converges to a random meridian connecting the opposite corners of the square. We also discuss analogous results for non-square Young tableaux.