  The time delays between the multiple images of a strong lens system, together
with a model of the lens mass distribution, allow a one-step measurement of a
cosmological distance, namely, the "time-delay distance" of the lens (D_dt)
that encodes cosmological information. The time-delay distance depends
sensitively on the radial profile slope of the lens mass distribution;
consequently, the lens slope must be accurately constrained for cosmological
studies. We show that the slope cannot be constrained in two-image systems with
single-component compact sources, whereas it can be constrained in systems with
two-component sources provided the separation between the image components can
be measured with milliarcsecond precisions, which is not feasible in most
systems. In contrast, we demonstrate that spatially extended images of the
source galaxy in two-image systems break the radial slope degeneracy and allow
D_dt to be measured with uncertainties of a few percent. Deep and
high-resolution imaging of the lens systems are needed to reveal the extended
arcs, and stable point spread functions are required for our lens modelling
technique. Two-image systems, no longer plagued by the radial profile slope
degeneracy, would augment the sample of useful time-delay lenses by a factor of
~6, providing substantial advances for cosmological studies.
