two identical charged balls are suspended from the samepoint by two massless identical thread of density of each ball is rho if system is immersed in a medium of density sigma ball remain undeflected then dielectric constant of medium is?

Ans is option 1. Assume that they were hanging while making some angle @ with the vertical. Find tan@ by balancing forces. Now this @ does not changes according to question, so by introducing bouyant force and change in electric force because of dielectric, again balance forces and find tan@. Equate these tan@s and you will get option 1 as answer.