In this paper, we study the existence and asymptotic behavior of solutions for a class of degenerate elliptic equation involving Grushin-type operator and Hardy potentials where −(Δx+|x|2αΔy) is the Grushin-type operator, α>0,2∗(s)=2(Q−s)Q−2 is the critical Sobolev-Hardy exponent and Q=m+(α+1)n is the homogenous dimension for Grushin operator. If 0≤μ<(Q−22)2,0<s<2, we will prove the existence of nontrivial, nonnegative solutions for this degenerate problem, and give the asymptotic behavior of solutions, at the singularity and at infinity.