直线OC:y=x√2/2
直线AB:x+y-1=0
上式联解得到C点坐标
x+(√2/2)x-1=0
xC=2/(2+√2),yC=√2/(2+√2)
C[2/(2+√2),√2/(2+√2)]
xA+xB=2xC=4/(2+√2),yA+yB=2yC=2√2/(2+√2)
由直线x+y-1=0
得y=1-x,带入椭圆方程
ax²+by²=1
ax²+b(1-x)²=1
(a+b)x²-2bx+b-1=0
xA+xB=2b/(a+b)=4/(2+√2),b=(√2)a
xA*xB=(b-1)/(a+b)=(√2a-1)/(a+√2a)
(xA+xB)²=8/(3+2√2)
(xA-xB)²=(yA-yBV)²=(xA+xB)²-4xA×xB=8/(3+2√2)-4(√2a-1)/(a+√2a)
(xA-xB)²+(yA-yB)²=AB²
2[8/(3+2√2)-4(√2a-1)/(a+√2a)]=(2√2)²=8
2/(3+2√2)-(√2a-1)/(a+√2a)=1
2/(3+2√2)-1=(√2a-1)/(a+√2a)
(-1-2√2)/(3+2√2)=(√2a-1)/(a+√2a)
a=1/3
b=(√2)a=√2/3
椭圆方程为x²/3+√2y²/3=1