Analysing 2-$(v,k,2)$ designs admitting a flag-transitive almost simple automorphism group with socle $PSL(2,q)$ by means of conics and hyperovals of $PG(2,q)$

The classification of the $2$-designs with λ=2 admitting a flag-transitive automorphism groups with socle $PSL(2,q)$ is completed by settling the two open cases in {ABDT}. The result is achieved by using conics and hyperovals of $PG(2,q)$.