We prove strong existence and pathwise uniqueness for two stochastic models of a seeker steering toward a target, written in polar coordinates. In both, the angular drift carries a $\frac1{r}$-type singularity which belongs to the supercritical regime in $\mathbb{R}^2$. Standard results for SDEs with singular drift therefore do not apply, and we give a new proof of strong well-posedness based on a pathwise argument. The two models arise from sailboat navigation and proportional navigation. We study the limiting regime in which the stopping radius around the target tends to zero and prove that, despite the singularity at the origin, each system admits a unique strong solution up to the hitting time of the target. These results provide an example of strong well-posedness in a regime where the general theory does not apply.