(1)一个顶点为(2,0),则a=2
离心率为e=c/a=√3/2,则c=√3/2*a=√3
∴b²=a²-c²=4-3=1
∴椭圆方程为x²/4+y²=1
(2)易知左右焦点分别为F1(-√3,0),F2(√3,0)
过F1且斜率为1的直线方程为y=x+√3
直线交椭圆于A,B两点,易知
S△ABF2=S△AF1F2+S△BF1F2
=1/2*|F1F2|*|y(A)|+1/2*|F1F2|*|y(B)|
=1/2*|F1F2|*[|y(A)|+|y(B)|]
=1/2*|F1F2|*|y1-y2|
将直线x=y-√3代入椭圆可得
(y-√3)²/4+y²=1,整理可得
5y²-2√3y-1=0
由韦达定理可得
y1+y2=2√3/5,y1y2=-1/5
∴|y1-y2|=√(y1-y2)²
=√[(y1+y2)²-4y1y2]
=√[(2√3/5)²-4*(-1/5)]
=4√2/5
∴S△ABF2=1/2*|F1F2|*|y1-y2|
=1/2*2√3*4√2/5
=4√6/5
∴△ABF2的面积为4√6/5