= Solution
For an <affine parameter> $\lambda$, the <geodesic equation> and the connection coefficients from part (a) give
$$
\ddot r-\sinh r\cosh r\,\dot\phi^2=0,
\qquad
\ddot\phi+2\coth r\,\dot r\dot\phi=0,
$$
where dots denote derivatives with respect to $\lambda$.
On a line of constant $\phi$, one has $\dot\phi=0$. The equations reduce to $\ddot r=0$, so $r$ itself can be chosen as an affine parameter. Every such radial line is therefore a <geodesic>.
For a circle $r=r_0>0$, the first equation instead requires
$$
\sinh r_0\cosh r_0\,\dot\phi^2=0.
$$
Both hyperbolic factors are positive when $r_0>0$, so $\dot\phi=0$. This gives only a constant point, not a parametrized circle. Thus no circle of constant positive $r$ is a geodesic, in agreement with the <coordinate geodesics of the hyperbolic polar metric>.
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