  Lensed quasars and supernovae can be used to study galaxies' gravitational
potential and measure cosmological parameters. The typical image separation of
objects lensed by galaxies is of the order of 0.5". Therefore, finding the ones
with small separations, and measuring their time-delays using ground-based
observations is challenging. We suggest a new method to identify lensed quasars
and simultaneously measure their time-delays, using seeing-limited synoptic
observations in which the lensed quasar images and the lensing galaxy are
unresolved. We show that using the light curve of the combined flux, and the
astrometric measurements of the center-of-light position of the lensed images,
the lensed nature of a quasar can be identified, and its time-delay can be
measured. We provide the analytic formalism to do so, taking into account the
measurement errors and the fact that the power spectra of quasar light curves
is red (i.e., the light curve is highly correlated). We demonstrate our method
on simulated data, while its implementation to real data will be presented in
future papers. Our simulations suggest that, under reasonable assumptions, the
new method can detect unresolved lensed quasars and measure their time delays,
even when the image separation is below 0.1", or the flux ratio between the
faintest and brightest images is as low as 0.03. Python and MATLAB
implementations are provided. In a companion paper, we present a method for
measuring the time delay using the combined flux observations. Although the
flux-only method is less powerful, it may be useful in cases in which the
astrometric information is not relevant (e.g., reverberation mapping).
