Diffusion processes with boundaries are models of transport phenomena with wide applicability across many fields. These processes are described by their probability density functions (PDFs), which often obey Fokker-Planck equations (FPEs). While obtaining analytical solutions is often possible in the absence of boundaries, obtaining closed-form solutions to the FPE is more challenging once absorbing boundaries are present. As a result, analyses of these processes have largely relied on approximations or direct simulations. In this paper, we studied two-dimensional, time-homogeneous, spatially-correlated diffusion with linear, axis-aligned, absorbing boundaries. Our main result is the explicit construction of a full family of closed-form solutions for their PDFs using the method of images (MoI). We found that such solutions can be built if and only if the correlation coefficient $ρ$ between the two diffusing processes takes one of a numerable set of values. Using a geometric argument, we derived the complete set of $ρ$'s where such solutions can be found. Solvable $ρ$'s are given by $ρ= - \cos \left( \fracπ{k} \right)$, where $k \in \mathbb{Z}^+ \cup \{ +\infty\}$. Solutions were validated in simulations. Qualitative behaviors of the process appear to vary smoothly over $ρ$, allowing extrapolation from our solutions to cases with unsolvable $ρ$'s.