(1)设A（x1,y1),B(x2,y2)
e=√3/3 c=1a=c/e=√3b=√2
椭圆方程是x^2/3+y^2/2=1
联立x^2/3+y^2/2=1
y=-x+1
消去y,得
5x^2-6x-3=0
｜AB｜=√(1+1)｜x1-x2｜
x1+x2=6/5
x1x2=-3/5
(x1-x2)^2=(x1+x2)^2-4x1x2=96/25
｜AB｜=x1x2+y1y2=0
｜AB｜=√2√(96/25)=(8√3 ) /5
(2)设A（x1,y1),B(x2,y2)
联立x^2/a^2+y^2/b^2=1
y=-x+1
消去y,得
(a^2+b^2)x^2-2a^2+a^2-a^2b^2=0
有 x1+x2=2a^2/(a^2+b^2)①
x1x2=(a^2-a^2b^2)/(a^2+b^2)②
向量OA与向量OB互相垂直
向量OA*向量OB=x1x2+y1y2=0
y1=-x1+1,y2=-x2+1,代入x1x2+y1y2=0
2x1x2-(x1+x2)+1=0③
把①②代入③得
a^2+b^2=2a^2b^2
b^2=a^2-c^2,c=ae代入上式得
a^2=2(1-e^2)/(2-e^2)④
当1/2