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Rotate the disc so that $P=r\in (0,1)$ lies on the positive real axis. From part (b), the perpendicular hyperbolic line $\ell$ is a Euclidean circle orthogonal to the unit circle, with centre $c>1$ on the real axis and radius $R=c-r$. Orthogonality of the two circles givesCombining the two equations yields

The radial distance formula from part (b) saysThe hyperbolic double-angle identities now giveThereforeThis is the Euclidean circle representing a perpendicular hyperbolic line.