  The statistics of first-passage times of random walks to target sites has
proved to play a key role in determining the kinetics of space exploration in
various contexts. In parallel, the number of distinct sites visited by a random
walker and related observables have been introduced to characterize the
geometry of space exploration. Here, we address the question of the joint
distribution of the first-passage time to a target and the number of distinct
sites visited when the target is reached, which fully quantifies the coupling
between kinetics and geometry of search trajectories. Focusing on 1-dimensional
systems, we present a general method and derive explicit expressions of this
joint distribution for several representative examples of Markovian search
processes. In addition, we obtain a general scaling form, which holds also for
non Markovian processes and captures the general dependence of the joint
distribution on its space and time variables. We argue that the joint
distribution has important applications to various problems, such as a
conditional form of the Rosenstock trapping model, and the persistence
properties of self-interacting random walks.
