= Solution
The conditions are singularities of the image-plane <Jacobian matrix>, where the formal <lensing magnification> of a <point-like astronomical source> diverges. A small source is strongly stretched in the corresponding <eigenvector> direction: tangentially near $\theta_t$, radially near $\theta_r$. Their mapped loci are source-plane <gravitational-lensing caustics>, which determine where image counts change.
The tangential circle maps to $\beta=0$ and gives the aligned <Einstein ring>. To find the radial caustic, put $R=\theta_E/\theta_c>1$. At $\theta_r$, $\sqrt{\theta_r^2+\theta_c^2}=\theta_cR^{1/3}$ and $\theta_r=\theta_c\sqrt{R^{2/3}-1}$. Substitution in the signed lens mapping gives a negative source position for positive $\theta_r$; its magnitude is
$$
\boxed{\beta_c=\theta_c(R^{2/3}-1)^{3/2}.}
$$
An offset source with $0<|\beta|<\beta_c$ has three images. For $\beta>0$, one outer image is on the positive side and two are on the negative side, as the sketch shows. The central and outer images have positive parity; the intermediate image has negative parity. On the radial caustic the inner pair merges, with a separate outer image remaining; outside it only the outer image survives. At exact alignment a central image coexists with the ring. Finite source extent smooths the formal divergences. The singular zero-core limit removes the central image, so its image-count rule differs from the finite-core case.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2006/iii/paper-72-cored-lens.png]
{title=Three images inside the radial caustic of a softened isothermal lens and its radial and tangential critical radii}
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