Question

已知椭圆x^2/a^2+y^2/b^2=1(a>0,b>0)的离心率e=√6/3,右焦点F到直线x/a+y/b=0的距离

Answer 1

(2)c=2,F(2,0),

设AB:y=kx+m,代入x^2/6+y^2/2=1得

x^2+3(k^2x^2+2kmx+m^2)=6,

整理得(1+3k^2)x^2+6kmx+3m^2-6=0,

设A(x1,y1),B(x2,y2),则x1+x2=-6km/(1+3k^2),x1x2=(3m^2-6)/(1+3k^2),

∠AFM=∠BFN,

∴AF,BF的斜率之和=y1/(x1-2)+y2/(x2-2)=0,

∴y1(x2-2)+y2(x1-2)=(kx1+m)(x2-2)+(kx2+m)(x1-2)=2kx1x2+(m-2k)(x1+x2)-4m=0,

∴2k(3m^2-6)-6km(m-2k)-4m(1+3k^2)=0,

∴6km^2-12k-6km^2+12k^2*m-4m-12k^2*m=0,

∴m=-3k,

∴AB:y=kx-3k,过定点(3,0).