x^2-3y^2=3
x^2-3(k^2x^2+2mkx+m^2)=3
(3k^2-1)x^2+6mkx+3m^2+3=0
x有2解,所以△>0
△=36m^2k^2-36m^2k^2+12m^2-36k^2+12
=12(m^2-3k^2+1)>0
m^2-3k^2+1>0
根据韦达定理,x1+x2=-6mk/(3k^2-1)
所以 y1+y2=kx1+m+kx2+m
=k(x1+x2)+2m
=-6mk^2/(3k^2-1)+2m
=(-6mk^2+6mk^2-2m)/(3k^2-1)
=-2m/(3k^2-1)
所以MN中点(-3mk/(3k^2-1),-m/(3k^2-1)
MN斜率为k
MN中垂线斜率为-1/k
MN中垂线方程：y+m/(3k^2-1)=(-1/k)[x+3mk/(3k^2-1)]
代入A(0,-1)
-1+m/(3k^2-1)=-3m/(3k^2-1)
4m/(3k^2-1)=1
m=(3k^2-1)/4≠0,k≠±√3/3
△=m^2-3k^2+1=m^2-4m>0
m>4 或者 m