(a) Describe the Poincaré disc model $D$ for the hyperbolic plane by giving the appropriate Riemannian metric.

(b) Let $a \in D$ be some point. Write down an isometry $f: D \rightarrow D$ with $f(a)=0$.

(c) Using the Poincaré disc model, calculate the distance from 0 to re $e^{i \theta}$ with $0 \leqslant r<1$

(d) Using the Poincaré disc model, calculate the area of a disc centred at a point $a \in D$ and of hyperbolic radius $\rho>0$.