We assume that, when having the choice between only two places to jump to, the grasshopper never jumps back to the point from which he got to that place. Let us denote by $P_1$ the starting point of the grasshopper, with $P_2$ the point on the line on which he has jumped from $P_1$, and so on. As the length of the jumps are all equal to $r$, $OP_1P_2P_3$ is a rhombus (possibly a degenerate one). Similarly, $OP_3P_4P_5$ is also a rhombus. It follows that the triangles $P_1OP_5$ and $P_2P_3P_4$ are congruent (SAS), and from here we obtain that $P_1P_5$ is parallel to $\ell$. We deduce that $P_5$ is the reflection of $P_1$ across the perpendicular line from $O$ onto $\ell$. (This fact remains true even in the degenerate cases.) From $P_5$, the grasshopper can get to $P_9$ which, as above, is the reflection of $P_5$ across the perpendicular line from $O$ onto $\ell$, i.e. $P_1$. In conclusion, the grasshopper can reach only the points $P_k$, $k = \overline{1,8}$ (which are not necessarily distinct).