把y=1-x代入ax²+by²=1得
ax^+b(1-2x+x^)=1,
(a+b)x^-2bx+b-1=0,
设A(x1,y1),B(x2,y2),则
x1+x2=2b/(a+b),x1x2=(b-1)/(a+b),
AB中点M(b/(a+b),a/(a+b)),直线OM的斜率为a/b=2,a=2b.
由OA⊥OB得0=x1x2+y1y2=x1x2+(1-x1)(1-x2)=2x1x2-(x1+x2)+1
=[2(b-1)-b+(a+b)]/(a+b)=(a+2b-2)/(a+b),
解得a=1,b=1/2.
∴椭圆方程是x^+y^/2=1.