已知ABC均在椭圆M:x^2/a^2+y^2=1(a>1)上,直线AB,AC分别是椭圆的左右焦点F1,F2,当向量
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①当向量AC·向量F1F2=0时,AF2垂直于F1F2,
9向量AF1·向量AF2
=9|AF1||AF2|cosA=9|AF2|^2=|AF1|^2
=>|AF1|=3|AF2| 又|AF1|+|AF2|=2a
=>|AF1|=3a/2,|AF2|=a/2,2c=|F1F2|=(√2)a
=>a^2=2(a^2-2)=>a^2=4
椭圆M的方程为x^2/4+y^2/2=1
②设P,E,F的坐标依次为(2cosα,(√2)sinα),(cosβ,2+sinβ),(-cosβ,2-sinβ)
则向量PE·向量PF
=(cosβ-2cosα)(-cosβ-2cosα)+
(2+sinβ-(√2)sinα)(2-sinβ-(√2)sinα)
=4(cosα)^2-4(√2)sinα+2(sinα)^2+3
=-2(sinα)^2-4(√2)sinα+7
=11-2(sinα+√2)^2
当sinα=-1时,向量PE·向量PF取最大值5+4√2
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