For a rational two-dimensional nonlinear in parameters model used in analytical chemistry, we investigate how homothetic transformations of the design space affect the number of support points in the optimal designs. We show that there exist two types of optimal designs: a saturated design (i.e. a design with the number of support points which is equal to the number of parameters) and an excess design (i.e. a design with the number of support points which is greater than the number of parameters). The optimal saturated designs are constructed explicitly. Numerical methods for constructing optimal excess designs are used.