Write down the Riemannian metric in the half-plane model of the hyperbolic plane. Show that Möbius transformations mapping the upper half-plane to itself are isometries of this model.

Calculate the hyperbolic distance from $i b$ to $i c$, where $b$ and $c$ are positive real numbers. Assuming that the hyperbolic circle with centre $i b$ and radius $r$ is a Euclidean circle, find its Euclidean centre and radius.

Suppose that $a$ and $b$ are positive real numbers for which the points $i b$ and $a+i b$ of the upper half-plane are such that the hyperbolic distance between them coincides with the Euclidean distance. Obtain an expression for $b$ as a function of $a$. Hence show that, for any $b$ with $0<b<1$, there is a unique positive value of $a$ such that the hyperbolic distance between $i b$ and $a+i b$ coincides with the Euclidean distance.