以AB连线为x轴,AB,中点为原点O建立坐标系
设A(-m,0),B(m,0),M(x,y) 则N(x,0)
∵MB^2=λAN●NB
∴(x-m)²+y²=λ(x+m)(m-x)
∴x²-2mx+m²+y²=λm²-λx²
(1+λ)x²+y²-2mx=λm²
当 λ=-1时,y²=2mx-m²
即y²=2m(x-m/2)为抛物线
λ=0时,(x-m)²+y²=0,轨迹是定点(m,0)
当 λ-1x²-2mx+m²+y²=λm²-λx²(1+λ)x²+y²-2mx=λm²这个漏了个m^2吧对的，不好意思疏漏 (1+λ)x²+y²-2mx=(λ-1)m²当 λ=-1时，y²=2mx-2m²即y²=2m(x-m)为抛物线λ=0时，(x-m)²+y²=0,轨迹是定点(m,0) 当 λ-1<λ<0或λ>0时， (1+λ)x²+y²-2mx=(λ-1)m²(1+λ)[x²-2m/(1+λ)*x+m²/(1+λ)²]+y²=(λ-1)m²+m²/(1+λ) [x-m/(1+λ)]²+y²/(1+λ)=m²λ²/(1+λ)²为椭圆当λ<-1时， [x-m/(1+λ)]²+y²/(1+λ)=m²λ²/(1+λ)²为双曲线