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In this paper, we consider the problem of planning optimal paths for a differential-drive robot with limited sensing, that must maintain visibility of a fixed landmark as it navigates in its environment. In particular, we assume that the robot's vision sensor has a limited field of view (FOV), and that the fixed landmark must remain within the FOV throughout the robot's motion. We first investigate the nature of extremal paths that satisfy the FOV constraint. These extremal paths saturate the camera pan angle. We then show that optimal paths are composed of straight-line segments and sections of these these extremal paths. We provide the complete characterization of the shortest paths for the system by partitioning the plane into a set of disjoint regions, such that the structure of the optimal path is invariant over the individual regions