I will talk about a principal eigenvalue problem for a second-order elliptic operator with a very small diffusion coefficient in one direction. In this regime, how do the principal eigenvalue and the principal eigenfunction behave? To handle the interaction between the slow and fast variables, I will use a representation of the principal eigenfunction as a quasi-stationary distribution of a killed process. Next, in some particular cases, I will give the limit of the principal eigenvalue when the diffusion coefficient converges to zero in one direction.