(1)記A,B橫坐標(biāo)分別為x1,x2
向量AF=2向量FB
由定比分點(diǎn)得(x1+2x2)/3=-c
即x1+2x2=-3c
由第二定義可得|AF|/(x1+a²/c)=c/a
則焦半徑|AF|=cx1/a+a
同理|BF|=cx2/a+a
|AF|+2|BF|=2|AF|=c(x1+2x2)/a+3a
則|AF|=-3c^2/2a+3a/2 ①
由|AF|=2|BF|得x1-2x2=a^2/c ②
根據(jù)圖像分析得
x2+a²/c=a²/c-c-|BF|cos60°
x1+a²/c=a²/c-c+|AF|cos60°
x2=-c-|AF|/4
x1=-c+|AF|/2
①②==>x1-2x2=c+|AF|=c-3c^2/2a+3a/2=a^2/c
1-3e/2+3/2e=1/e^2==>1-1/e^2=3(e-1/e)/2
==>(e^2-1)/e^2=3(e^2-1)/2e得e=2/3
(2)c=2a/3
|AB|=|AF|+|BF|=3|AF|/2=-9c^2/4a+9a/4
=-a+9a/4=5a/4=15/4 得a=3
橢圓C的方程為x^2/9+y^2/5=1