1)
x²/9+y²=1
则a=3, b=1, c=√(9-1)=2√2
左焦点为F1(-2√2,0)
直线为y=√3/3(x+2√2)
代入椭圆得:
x²/9+1/3(x²+8+4√2x)=1
得:4x²+12√2x+15=0
x1+x2=-3√2, x1x2=15/4
|AB|²=(x1-x2)²+(y1-y2)²=(x1-x2)²+1/3(x1-x2)²=4/3[(x1+x2)²-4x2x2]=4/3[18-15]=4
故|AB|=2
2) AB中点坐标为M((x1+x2)/2, (y1+y2)/2), 即:(-3√2/2, √6/6)
|F1M|²=(-2√2+3√2/2)²+(√6/6)²=1/2+1/6=2/3
|F1M|=√(2/3)=√6/3