设M(x1,y1),N(x2,y2)
OM⊥ON
∴x1x2+y1y2=0
联立y=k(x+√3),11x²+y²=9
11x²+k²(x+√3)²=9
(11+k²)x²+2√3k²x+3k²-9=0
△=(2√3k²)²-4(11+k²)(3k²-9)＞0
根据韦达定理有
x1x2=(3k²-9)/(11+k²)
y1y2=k(x1+√3)*k(x2+√3)=k²[x1x2+√3(x1+x2)+3]
=k²[(3k²-9)/(11+k²)-√3*2√3k²/(11+k²)+3]
=-x1x2=(9-3k²)/(11+k²)
k²=1/3满足△＞0
k=±√3/3