In this paper, we study the location and stability of the collinear Lagrangian points for the RTBP case in which one of the primary bodies radiates and the other is oblate. We consider the effect of Poynting–Roberson drag and investigate how the location and stability of the Lagrangian points change with changes in the radiation parameter \(\beta \) and oblateness a. We apply our results to ten exoplanet systems: CoRoT-2 b, TOI-1278 b, HAT-P-20 b, Kepler-75 b, WASP-89 b, TIC 172900988 b, NGTS 9 b, LP 714-47 b, WASP-162 b and XO-3 b, data of which has been taken from NASA exoplanet archives, to find locations in these exoplanet systems where one can detect asteroids, primordial material or seeds where planet formation can take place. We find that the location of the collinear Lagrangian points changes with variations in radiation pressure and oblateness. Further, for all the ten planetary systems studied in this paper, the Lagrangian points are unstable and can be locations where we expect to find minor planets, asteroids or debris. The unstability of the Lagrangian points can be a possible cause of relocation and migration of planetesimals.