The conjugated base of strong acid is weak and the conjugated base of weak acid is strong. It can be explain—clearly by reversible reaction.
HB ↔ H+ + B–
Acid proton base
HB is strong acid, for that the equilibrium of the reaction to be right. So backward reaction (the addition of proton with B–) become less. So B– (conjugated base) become weak. On the other hand if the base is strong, the equilibrium of the reaction to be left. So the dissociation of acid becomes less and the acid will be weak.