汗!打的好辛苦的!
设A(x1,y1),B(x2,y2),M((x1+x2)/2,(y1+y2)/2).
由{x+y=1};{ax2+by2=1}∴(a+b)x2-2bx+b-1=0.
∴(x1+x2)/2=a/a+b.
∴M(b/a+b,a/a+b).
∵kOM=根号2/2,∴b=根号2a①
∴OA⊥OB,∴y1/x1•y2/x2=-1.
∴x1x2+y1y2=0.
∵x1x2=b-1/a+b,y1y2=(1-x1)(1-x2).
∴y1y2=1-(x1+x2)+x1x2=a-1/a+b,
∴(b-1)/(a+b)+(a-1)/(a+b)=0,∴a+b=2②
由①②得:a=2(根号2-1),b=2根号2(根号2-1).
∴所求方程为:2(根号2-1)x2+2根号2(根号2-1)y2=1.