c/a=√2/2,a=√2c,b^2=a^2-c^2=c^2,a^2=2b^2
x^2+2y^2=2b^2
y=x+1与x^2+2y^2=2b^2联立消去y得：
x^2+2(x+1)^2-2b^2=0
即3x^2+4x+2-2b^2=0
A(x1,y1),B(x2,y2)
x1+x2=-4/3,x1x2=(2-2b^2)/3
|AB|=√2√[(x1+x2)^2-4x1x2]
=√2√[16/9-8/3(1-b^2)]
O到AB的距离d=1/√2
SΔAOB=1/2 |AB| d
=√[16/9-8/3(1-b^2)]/2
由√[16/9-8/3(1-b^2)]/2=2/3
解得：b^2=1
此椭圆的方程为x^2/2+y^2=1