椭圆E:x^2/a^2+y^2/b^2=1过点P(3,1)其左右焦点分别为F1F2且向量F1P*向量F2P=-6

1
F1(-c,0) F2(c,0)
F1P=(3-c,1)
F2P=(3+c,1)
F1P*F2P=(3-c)(3+c)+1*1=-6
10-c^2=-6
c^2=16
c=4
9/a^2+1/(a^2-16)=1
9*(a^2-16)+a^2=a^2(a^2-16)
a^4-26a^2+144=0
a^2=18
b^2=2
方程x^2/18+y^2/2=1
2
M(5,y1)N(5,y2)y1>0>y2
F1M垂直F2Ny1/(5+4)=-1/[y2/(5-4)]
y1/9=-1/y2
y1y2=-9
MN=|y1-y2|=y1+9/y1=(√y1-3/√y1)^2+6
√y1=3/√y1,MN最小=6,S最小 y1=3 y2=-3
MN中点C,Cx=5,Cy=(3-3)/2=0
圆C方程：(x-5)^2+y^2=(6/2)^2=9