Question

已知椭圆E：X^2/a^2+y^2/b^2=1过点P(3,1),其左右焦点为F1,F2,且向量F1P*F2P=-6

Answer 1

(1)

F1P(3+c,1),F2P(3-c,1)

9-c^2+1=-6

c=4

9/a^2+1/(a^2-16)=1

解得:a^2=18或8(舍)

x^2/18+y^2/2=1

(2)设M点坐标为(5,m),N点坐标为(5,n)

F1M(9,m); F2N(1,n)

F1M*F2N=0

mn=-9

圆的方程为:(x-5)^2+[y-(m+n)/2]^2=(m-n)^2/4

即:x^2-10x+25+y^2+(m+n)y+mn=0

因为mn=-9,方程为:x^2-10x+16+y^2+(m+n)y=0

若x^2-10x+16=0并且y=0,该等式恒成立.

所以圆过定点(2,0)和(8,0)