the question u ve given can be under stood in 2 ways either the dispalcement can along the line joining the charges (or) the displacement is perpendicular to this linear system
if considering the two cases and solving i think the question is about the first case as the answer is coinciding and even the middle particle wont be having frequency for the second case any how you will be getting the proofs of both of them in the next post
please have mercy on me if i were wrong and correct me