B.根号2
已知椭圆x^2/a^2 +y^2/b^2=1(a>b>0）的离心率为根号3/2,设椭圆方程x^2/4+y^2=1
设直线方程为x=my+根号3 代入椭圆方程得
(m^2+4)y^2+2根号3y-1=0
若AF=3FB
所以|yA|=3|yB|
解方程(m^2+4)y^2+2根号3y-1=0
y=[-2根号3m±√(12m^2+4(m^2+4))] /[2(m^2+4)]
m>0 yA=[-2根号3m-√(12m^2+4(m^2+4))] /[2(m^2+4)]
yB=[-2根号3m+√(12m^2+4(m^2+4))] /[2(m^2+4)]
|yA|=3|yB|
|[-2根号3m-√(12m^2+4(m^2+4))] /[2(m^2+4)]|=3|[-2根号3m+√(12m^2+4(m^2+4))] /[2(m^2+4)]
2根号3m+√(12m^2+4(m^2+4))=3 √[12m^2+4(m^2+4))] -6根号3m
8根号3m=2√(16m^2+16) 64*3*m^2=64(m^2+1) 2m^2=1
m=根号2/2 k=1/m=根号2