1、把y=1-x代入x²/a²+y²/b²=1,(a²+b²)x²-2a²x+a²-a²b²=0,x1+x2=2a²/(a²+b²),x1x2=(a²-a²b²)/(a²+b²),把x=1-y代入x²/a²+y²/b²=1,(a²+b²)y²-2b²y+b²-a²b²=0,y1+y2=2b²/(a²+b²),y1y2=(b²-a²b²)/(a²+b²),OA⊥OB,则y1y2/x1x2=-1,(b²-a²b²)/(a²-a²b²)=-1,a²+b²=2a²b²,直线x+y-1=0在x、y轴截距为1,∵a>b>0,且OA⊥OB,∴a＞1,0＜b＜1,椭圆E与圆x²+y²=1的交点,x=±√[(a²-a²b²)/(a²-b²)],y=±√[(a²b²-b²)/(a²-b²)]；
2、(x1-x2)²=4a^4/(a²+b²)²-4(a²-a²b²)/(a²+b²),(y1-y2)²=4b^4/(a²+b²)²-4(b²-a²b²)/(a²+b²),(x1-x2)²+(y1-y2)²=(4a^4+4b^4)/(a²+b²)²-4(a²-a²b²+b²-a²b²)/(a²+b²)=(4a^4+4b^4)/(a²+b²)²=5/2,把b²=a²/(2a²-1)代入解得：a²=2或a²=2/3(舍去),b²=2/3,椭圆E的方程：x²/2+3y²/2=1.