In this paper, we consider the problem of locating $k$ obnoxious facilities of maximum radius, centered on a line segment $\overline{pq}$, amidst $n$ demand points in the plane so that none of the existing facility sites are affected. An $(1-\epsilon)$-approximation algorithm was given recently to solve the problem [CCCG 2021], where $\epsilon>0$. Here, we present two polynomial-time exact algorithms based on two different approaches: (i) the algorithm is based on doing a binary search on all candidate radii $L$ computed explicitly and runs in $O((nk)^2\log{(nk)}+(n+k)\log{(nk)})$ time, and (ii) the algorithm is based on Megiddo's parametric search and runs in $O((n+k)^2)$ time.