直线x+y-1=0
y=-x+1
代入ax²+by²=1
ax²+b(-x+1)²=1
ax²+bx²-2bx+b²-1=0
(a+b)x²-2bx+b²-1=0
设A,B的坐标分别为（xA,yA）（xB,yB）
xA+xB=2b/(a+b)
xA×xB=(b²-1)/(a+b)
点C的横坐标b/(a+b)
纵坐标(yA+yB)/2=(-xA+1-xB+1)/2=-(xA+xB-2)/2=-b/(a+b)+1=a/(a+b)
根据题意
[a/(a+b)]/[b(a+b)]=√2/2
a/b=√2/2
b=√2a(1)
AB=√(xA-xB)²+(yA-yB)²=√(xA-xB)²+(xA-xB)²=√2[(xA+xB)²-4xA×xB]
2√2=√2[4b²/(a+b)²-4(b²-1)/(a+b)]
1=b²/(a+b)²-(b²-1)/(a+b)
a²+2ab+b²=b²-(b²-1)(a+b)
a²+2ab+ab²+b²×b-a-b=0
b=√2a
a²+2ab+ab²+b²×√2a-a-√2a=0
a不为0
a+2√2a+2a²+2√2a²-1-√2=0
(2+2√2)a²+(2√2+1)a-(√2+1)=0
2a²+(3-√2)a-1=0
a=√2-1或a=-(√2+1)/2（舍去）
a=√2-1
b=2-√2
(√2-1)x²+(2-√2)y²=1
x²/(√2+1)+y²/(2+√2)/2=1