Question

从点A（6,8）向圆x^2+y^2=16任意引一割线L交圆于P1,P2两点,求弦P1P2的中点P的轨迹.

Answer 1

任意引一割线L:y-8=k(x-6),k=(y-8)/(x-6)

割线L交圆于P1,P2两点

k=(yP1-yP2)/(xP1-xP2)=(y-8)/(x-6)

P1P2的中点P(x,y)

xP1+xP2=2x,yP1+yP2=2y

x^2+y^2=16

(xP1)^2+(yP1)^2=16.(1)

(xP2)^2+(yP2)^2=16.(2)

(1)-(2):

(xP1+xP2)*(xP1-xP2)+(yP1+yP2)*(yP1-yP2)=0

(xP1+xP2)+(yP1+yP2)*(yP1-yP2)/(xP1-xP2)=0

2x+2y*(y-8)/(x-6)=0

P1P2的中点P的轨迹:

(x-3)^2+(y-4)^2=5^2