The paper presents a scheme to calculate approximations of reach sets and tubes for linear control systems with time-varying coefficients, bounds on the controls, and constraints on the state. The scheme provides tight external approximations by ellipsoid-valued tubes. The tubes touch the reach tubes from the outside at each point of their boundary so that the surface of the reach tube is totally covered by curves that belong to the approximating tubes. The result is an exact parametric representation of reach tubes through families of external ellipsoidal tubes. The parameters that characterize the approximating ellipsoids are solutions of ordinary differential equations with coefficients given partly in explicit analytical form and partly through the solution of a recursive optimization problem. The scheme combines the calculation of external approximations of infinite sums and intersections of ellipsoids, and suggests an approach to calculate reach sets of hybrid systems.