奥林匹克高手告诉你，高考不会这样出题的。

F1(-c,0),F2(c,0)

2AB=AF2+BF2

2AB+AF1+BF1=AF2+AF1+BF1+BF2=4a

3AB=4a

AB=4a/3

AB:y=x+c

b^2=a^2-c^2

x^2/a^2+(x+c)^2/(a^2-c^2)=1

(2a^2-c^2)x^2+2ca^2*x+2a^2*c^2-a^4=0

xA+xB=-2ca^2/(2a^2-c^2)

xA*xB=(2a^2*C^2-a^4)/(2a^2-c^2)

(xA-xB)^2=(yA-yB)^2=(xA+xB)^2-4xA*xB

AB^2=2(xA-xB)^2=2(xA+xB)^2-8xA*xB

(4a/3)^2=2*[-2ca^2/(2a^2-c^2)]^2-8*(2a^2*C^2-a^4)/(2a^2-c^2)

8c^4-14a^2*c^2+5a^4=0

8e^4-14e^2+5=0

(4e^2-5)*(2e^2-1)=0

e<1

e=1/√2