 Solution

Use the geodesic Lagrangian $L=\frac{1}{2}(y^2{\dot{x}}^2+x^2{\dot{y}}^2)$, where dots denote an affine parameter derivative. The Euler-Lagrange equations giveThese are the geodesic equations wherever $xy\ne 0$, so the Riemannian metric is nondegenerate. Substitution of $y=mx$, $\dot{y}=m\dot{x}$ and $\ddot{y}=m\ddot{x}$ makes both equations proportional toThus $\boxed{x^2=A\lambda +B,y=mx}$ is a family of affinely parametrized geodesics, on intervals staying within one quadrant. In particular, $m\ne 0$ and $x\ne 0$ keep the metric nonsingular; the coordinate axes themselves are not regular points of this metric. The coordinate $x$ need not be linear in the affine parameter, despite the straight coordinate-space shape of these curves.