The topic of orthogonal trajectories is taught as a geometric application of first order differential equations. Instructors usually elaborate on the concept of a family of curves to emphasize that they are different even if their members are of the same type. In this article the author considers five families of ellipses, discusses their characteristics, derives and then solves the differential equations to find their corresponding orthogonal trajectories. This material could be used as the basis of a student project in a differential equation course.