  In this paper, the containment problem for the defining ideal of a special
type of zero dimensional subschemes of $\mathbb{P}^2$, so called quasi star
configurations, is investigated. Some sharp bounds for the resurgence of these
types of ideals are given. As an application of this result, for every real
number $0 < \varepsilon < \frac{1}{2}$, we construct an infinite family of
homogeneous radical ideals of points in $\mathbb{K}[\mathbb{P}^2]$ such that
their resurgences lie in the interval $[2- \varepsilon ,2)$. Moreover, the
Castelnuovo-Mumford regularity of all ordinary powers of defining ideal of
quasi star configurations are determined. In particular, it is shown that, %the
defining ideal of a quasi star configuration, and all of them have linear
resolution.
