We show the insolvability of the Diophantine equation axd-y2-z2+xyz-b=0 in Z for fixed a and b such that a≡1(mod12) and b=2da-3, where d is an odd integer and is a multiple of 3. Further, we investigate the more general family with b=2da-3r, where r is a positive odd integer. As a consequence, we found an infinite family of hyperelliptic curves with trivial torsion over Q. We conclude by providing some numerical evidence corroborating the main results.