\noindent We address some direct and inverse problems, for the first-exit time (FET) $τ$ of a drifted Brownian motion with Poissonian resetting ${\cal X}(t)$ from an interval $(0,b)$ and the first-exit area (FEA) $A,$ namely the area swept out by ${\cal X}(t)$ till the time $τ$; this type of diffusion process ${\cal X}(t)$ is characterized by the fact that a reset to the position $x_R $ can occur according to a homogeneous Poisson process with rate $r>0.$ When the initial position ${\cal X}(0)= η\in (0,b)$ is deterministic and fixed, the direct FET problem consists in investigating the statistical properties of the FET $τ,$ whilst the direct FEA problem studies the probability distribution of the FEA $A$. The inverse FET problem regards the case when $η$ is randomly distributed in $(0,b)$ (while $r$ and $x_R $ are fixed); if $F(t)$ is a given distribution function on the time $t$ axis, the inverse FET problem consists in finding the density $g$ of $η,$ if it exists, such that $P[τ\le t ] = F(t), \ t >0.$ %In addition to the case of random initial position $η,$ we also study the case when the initial position $η$ and the resetting rate $r$ are fixed, whereas the reset position $x_R$ is random. Several explicit examples of solutions to the inverse FET problem are provided.