2010-27.md

course	course_year	question_number	tags	title	year
Geometry	IB	27	IB 2010 Geometry	Paper 3, Section II, F	2010

Describe the hyperbolic metric on the upper half-plane $H$. Show that any Möbius transformation that preserves $H$ is an isometry of this metric.

Suppose that $z_{1}, z_{2} \in H$ are distinct and that the hyperbolic line through $z_{1}$ and $z_{2}$ meets the real axis at $w_{1}, w_{2}$. Show that the hyperbolic distance $\rho\left(z_{1}, z_{2}\right)$ between $z_{1}$ and $z_{2}$ is given by $\rho\left(z_{1}, z_{2}\right)=\log r$, where $r$ is the cross-ratio of the four points $z_{1}, z_{2}, w_{1}, w_{2}$, taken in an appropriate order.