We obtain approximate solutions to the equations that govern the shape of giant unilamellar vesicles (GUVs) with two fluid phases. The equations involve a dimensionless small parameter related to the resistance to changes in its local mean curvature. Asymptotic solutions for the shape are obtained up to and including terms of first order in the small parameter. At this order, we determine a relationship between the tangent angle at the interface and the difference in the Gaussian curvature stiffnesses of the co-existing phases. This relationship demonstrates that a difference in the Gaussian curvature stiffnesses moves the phase boundary away from the neck, as determined in previous numerical studies. The analytical expression for the tangent angle obtained here can be used to determine elastic parameters for the membranes from experimental data. Use of the analytical expression will eliminate the need for the repeated generation of numerical solutions in the estimation of the material parameters. Our analytical solution also reduces the number of measurements needed as inputs for an existing boundary layer analysis.