Hyperbolic elliptic parabolic disks approximated by half distance bands

Hyperbolic elliptic parabolic disks can be described by the inequality $\frac{x^2}{C^2}+2y^2-2y\leq0$ ($0<C<1$) in the unit disk based Beltrami--Cayley--Klein model of the hyperbolic geometry, up to hyperbolic congruences. The hyperbolic elliptic parabolic disks considered above are sort of close to their supporting half distance bands given by the inequalities $\frac{x^2}{C^2}+ y^2-1\leq0$ and $y\geq0$. Here we consider what `close' might mean, and we look for even more precise approximations, in terms of area and circumference.