椭圆中心为原点o,焦点在x轴上,离心率e=根号2/2,直线y=x+1交椭圆于A,B两点,且三角形AOB的面积=2/3,

c/a=√2/2,a=√2c,b^2=a^2-c^2=c^2,a^2=2b^2x^2+2y^2=2b^2y=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=0A(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/√2SΔAOB=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