1.椭圆x^2/m+y^2/4＝1过抛物线y^2＝4x的焦点F(1,0),
∴m-4=1,m=5,
∴椭圆方程是x^2/5+y^2/4=1,①离心率=√5/5.
2.过F斜率为√2的直线：y=√2(x-1),
代入①*20,得4x^2+10(x^2-2x+1)=20,
整理得14x^2-20x-10=0,
化简得7x^2-10x-5=0,
△=100+140=240,
∴|AB|=√(3△)=12√5.
3.l:x=y/√2+m,②
代入①*20,得4(y^2/2+m√2y+m^2)+5y^2=20,
整理得7y^2+4√2my+4m^2-20=0,③
y=[-2√2m土√(140-20m^2)]/7,
∴S△AF1F2=(1/2)|F1F2|*|yA|=|-2√2m土√(140-20m^2)|/7,
设f(m)=-2√2m+√(140-20m^2),-√70,f(m)是增函数,m>-√2时f'(m)