①當(dāng)向量AC·向量F1F2=0時(shí),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
橢圓M的方程為x^2/4+y^2/2=1
②設(shè)P,E,F的坐標(biāo)依次為(2cosα,(√2)sinα),(cosβ,2+sinβ),(-cosβ,2-sinβ)
則向量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
當(dāng)sinα=-1時(shí),向量PE·向量PF取最大值5+4√2