Motivated by encouraging numerical results in the literature, in this note we consider two specific variants of the trust-region subproblem and provide exact semidefinite representable reformulations. The first is over the intersection of two balls; the second is over the intersection of a ball and a special second-order conic representable set. Different from the technique developed in the literature, the reformulations in this note are based on partitions of the feasible regions into sub-regions with known lifted convex hulls.