  Balanced pairs appear naturally in the realm of Relative Homological Algebra
associated to the balance of right derived functors of the $\mathsf{Hom}$
functor. A natural source to get such pairs is by means of cotorsion triplets.
In this paper we study the connection between balanced pairs and cotorsion
triplets by using recent quiver representation techniques. In doing so, we find
a new characterization of abelian categories having enough projectives and
injectives in terms of the existence of complete hereditary cotorsion triplets.
We also give a short proof of the lack of balance for derived functors of
$\mathsf{Hom}$ computed by using flat resolutions which extends the one showed
by Enochs in the commutative case.
