1）c=2,离心率e=√6/3=c/a,∴a^2=6,b^2=2,椭圆方程为(x^2/6) + (y^2/2) = 1
2）∵3=6/2 = a^2/c,因此x=3恰好是椭圆的右准线,过A、B作x=3的垂线段,垂足E、F
按椭圆定义：AB = FA+FB=AE·e + BF·e,设A(x1,y1),B(x2,y2)
则AB=e·(6-x1-x2),设AB中点Q(x0,y0),则x1+x2=2x0,y1+y2=2y0
∵直线AB斜率存在,∴Q不会落在坐标轴上∴x0≠0,y0≠0
把A、B坐标代入椭圆方程并相减：[(x2)^2 - (x1)^2]/6 = -[(y2)^2 - (y1)^2]
∴6(y2 - y1)/(x2 - x1) = -(x2+x1)/(y2+y1),又∵直线AB斜率为k,∴6k=-x0/y0
∵△ABP是等边三角形,∴PQ垂直平分AB,且PQ=(√3/2)AB
而k(PQ)=-1/k(AB) = -1/k ∴过P、Q的直线为：y - y0 = (-1/k)(x - x0),∵P在x=3上,
∴求得P纵坐标=y0+(6y0/x0)(3 - x0) = y0 - 6y0 + (18y0/x0) = y0[(18/x0) - 5]
PQ^2=(3-x0)^2 + 36(y0)^2·[(3/x0) - 1]^2 = (3/4)(AB)^2 = (3/4)·(2/3)·4(3-x0)^2 = 2(3-x0)^2
因为P、Q不重合,∴x0≠3,∴1 + [36(y0)^2/(x0)^2] = 2,解得k = -x0/6y0 = -1或1
据此求得：斜率为k且经过焦点F(2,0)的直线l的方程为：x+y-2=0或x-y-2=0