= Solution
The <Cayley transform> $w=(z-i)/(z+i)$ maps the upper half-plane conformally onto the unit disc. Its inverse is $z=i(1+w)/(1-w)$, with
$$
|dz|^2=\frac{4|dw|^2}{|1-w|^4},\qquad\operatorname{Im}z=\frac{1-|w|^2}{|1-w|^2}.
$$
Transporting the given <hyperbolic metric> therefore gives the <Poincare disc model>
$$
\boxed{ds_D^2=\frac{4|dw|^2}{(1-|w|^2)^2}.}
$$
The <Cayley transform> is an <isometry> for this metric, so it carries the half-plane <geodesics> to disc <geodesics>. A <Möbius transformation> sends generalized circles to generalized circles and preserves angles. Hence these are exactly \b[diameters and arcs of circles meeting the unit circle orthogonally]. Multiplying this <Riemannian metric> by any positive constant leaves the unparametrized <geodesics> unchanged.
\Image[/past-exam-of-the-geometry-course-of-the-university-of-oxford/2010-hyperbolic-geodesics.png]
{title=Half-plane geodesics and their Cayley images in the disc}
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