Question

椭圆x²/25+y²/9=1上一点到两个焦点的距离之积为m,则m取最大值时,点P的坐标为多少?

Answer 1

a=5,b=3,c=4

PF1*PF2=M

PF1+PF2=2a=10

(PF1+PF2)^2=100

(PF1)^2+(PF2)^2=100-2PF1*PF2=100-2M

(PF1)^2+(PF2)^2≥2PF1*PF2

(PF1)^2+(PF2)^2≥2M

100-2M≥2M

M≤25

PF1=PF2=a=5,M取最大值=25

当M取最大值时,点P的坐标(0,±4)