设A(x1,y1),B(x2,y2).则x1,x2是x^2/a^2+y^2/b^2=1和X+Y=1联立方程的2根.
联立方程得:(a^2+b^2)x^2-2a^2x+a^2-a^2*b^2=0.
x1x2=(a^2-a^2*b^2)/(a^2+b^2),x1+x2=2a^2/(a^2+b^2).
OA垂直OB==>x1x2+y1y2=0,y=1-x.前式计算整理得(分子):a^2+b^2-2a^2*b^2=0==>b^2=a^2/(2a^2-1).
则x^2/a^2+y^2/b^2=1==>x^2/a^2+y^2(2-1/a^2)=1==>(1/a^2)*(x^2-y^2)+(2y^2-1)=0.令2y^2-1=0,x^2-y^2=0得x=y=(+,-)2^0.5/2.
5^0.5