Let a >1,b>1 and (α,b)=1,and let h(-αb)be the class number of the imaginaryquadratic field .In this paper,we have proved the following results.Theorem 1. If the positive integers x,y, n and k satisfyαx ̄2+by ̄2=4k ̄2, k>1,x|α,y|b,where symbol x|αmeans that α is divided exactly by each prime factor of x .Them hb)≡0(od 2n),except 5.5 ̄2+3.1 ̄2=4.2 ̄5,3.9 ̄2+13.1 ̄2=4.2 ̄6 and n=3 ,of by ̄2=3k+λ(λ=±1).Teorem 2.If the positive integers x,y,n and k satisfyαx ̄2+by ̄2=k ̄n,αb≡2(mod 4),x|α,y|b,then