Paper 3, Section I, F

(i) Write down the Poincaré metric on the unit disc model $D$ of the hyperbolic plane. Compute the hyperbolic distance $\rho$ from $(0,0)$ to $(r, 0)$, with $0<r<1$.

(ii) Given a point $P$ in $D$ and a hyperbolic line $L$ in $D$ with $P$ not on $L$, describe how the minimum distance from $P$ to $L$ is calculated. Justify your answer.

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