= Solution
The tangential condition $\bar\kappa=1$ gives $\sqrt{\theta_t^2+\theta_c^2}=\theta_E$, hence
$$
\boxed{\theta_t=\sqrt{\theta_E^2-\theta_c^2}.}
$$
This is a nonzero <gravitational-lensing critical curve> precisely when $\theta_E>\theta_c$. It is also the radius of the <Einstein ring> for an aligned source in this cored model, so the strength parameter $\theta_E$ itself is not the cored model's ring radius.
For completeness the radial condition is equally explicit. Subtracting the mean <lensing convergence> from twice the local one gives
$$
2\kappa-\bar\kappa=\frac{\theta_E\theta_c^2}{(\theta^2+\theta_c^2)^{3/2}}.
$$
Thus the <radial critical curve of an axisymmetric lens> has radius
$$
\boxed{\theta_r=\sqrt{(\theta_E\theta_c^2)^{2/3}-\theta_c^2},}
$$
again nonzero exactly when $\theta_E>\theta_c$.
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