A result of Lefebvre [Lefebvre, M. (2001). A different class of homing problems. Systems & Control Letters 42, 347-352] is here extended to a two-dimensional homing problem with a risk-sensitive cost criterion. It is shown that the optimal control is given explicitly, and moreover, the optimal value function has a simple probabilistic representation associated with a backward stochastic differential equation with a random terminal time. (C) 2009 Elsevier Ltd. All rights reserved.