椭圆ax^2+bx^2=1与直线y=1-x交于A,B两点,过原点与线段AB中点的直线斜率为根号3/2,求椭圆的离心率.

联立方程：ax^2+b(1-x)^2=1(a+b)x^2-2bx+b-1=0xA+xB=2b/(a+b)yA+yB=1-xA+1-xB=2-2b/(a+b)=2a/(a+b)AB中点(b/(a+b),a/(a+b))斜率k=a/(a+b)/[b/(a+b)]=a/b=V3/2e=c/a=V(a^2-b^2)/a=V(1-(b/a)^2)=1/2