  Recently, S. Li and A. Pott\cite{LP} proposed a new concept of intersection
distribution concerning the interaction between the graph
$\{(x,f(x))~|~x\in\F_{q}\}$ of $f$ and the lines in the classical affine plane
$AG(2,q)$. Later, G. Kyureghyan, et al.\cite{KLP} proceeded to consider the
next simplest case and derive the intersection distribution for all degree
three polynomials over $\F_{q}$ with $q$ both odd and even. They also proposed
several conjectures in \cite{KLP}.
  In this paper, we completely solve two conjectures in \cite{KLP}. Namely, we
prove two classes of power functions having intersection distribution:
$v_{0}(f)=\frac{q(q-1)}{3},~v_{1}(f)=\frac{q(q+1)}{2},~v_{2}(f)=0,~v_{3}(f)=\frac{q(q-1)}{6}$.
We mainly make use of the multivariate method and QM-equivalence on $2$-to-$1$
mappings. The key point of our proof is to consider the number of the solutions
of some low-degree equations.
