(1)由橢圓定義,c/a=|F1F2|/(|MF1|+|MF2|),
設(shè)∠MF1F2=α,因∠F1MF2=π/3,故0<α<2π/3,由正弦定理,
c/a=sin(π/3)/[sinα+sin(2π/3-α)]
=sin(π/3)/[2sin(π/3)cos(π/3-α)]
=1/[2cos(π/3-α)],
cos(π/3-α)的值域是(1/2,1],
∴離心率e=c/a的取值范圍是[1/2,1).
(2)e=1/2時(shí)a=2c,b=c√3,設(shè)P(2ccost,√3csint),則
PN^2=(2ccost)^2+(√3csint-3√3)^2
=4c^2[1-(sint)^2]+3c^2(sint)^2-18csint+27
=-c^2(sint)^2-18csint+4c^2+27
=-(csint+9)^2+4c^2+108,
|PN|<=4√3,∴sint=-1時(shí)PN^2取最大值3c^2+18c+27=48,
∴c^2+6c-7=0,c>0,
∴c=1,a=2,b=√3,橢圓的方程是x^2/4=y^2=1.
(3)當(dāng)P由上頂點(diǎn)向右頂點(diǎn)運(yùn)動(dòng)時(shí)|t|由0增大至(a-c)/a=1-e,由（1）,
t的取值范圍是(-1/2,1/2).