Suppose that $D$ is the unit disc, with Riemannian metric

$$d s^{2}=\frac{d x^{2}+d y^{2}}{1-\left(x^{2}+y^{2}\right)}$$

[Note that this is not a multiple of the Poincaré metric.] Show that the diameters of $D$ are, with appropriate parametrization, geodesics.

Show that distances between points in $D$ are bounded, but areas of regions in $D$ are unbounded.