Question

椭圆方程设P是椭圆X2/a2+Y2/b2(a大于b大于0)上任意一点（不是长轴端点）,F1,F2是焦点,三角形PF1F2

Answer 1

PI/BI=△F1IP/△F1IB(同高不同底)

=(F1P*r/2)/(F1B*r/2)=F1P/F1B

=F1P/(F1F2-F2B) .(1)

r为△F1F2P的内切圆半径

同理

PI/BI=△F2IP/△F2IB

=(F2P*r/2)/(F2B*r/2)=F2P/F2B .(2)

由(1),(2)式得

F1P/(F1F2-F2B)=F2P/F2B

F1P＊F2B＝F2P*F1F2-F2P*F2B.

(F1P+F2P)＊F2B＝F2P*F1F2 ...(3)

而椭圆

X^2/a^2+Y^2/b^2=1

其离心率

e=c/a ,

而 a^2=b^2+c^2

焦距 F1F2=2c,

而 F1P+F2P=2a

代入(3)

(F1P+F2P)＊F2B＝F2P*F1F2

2a*F2B=F2P*2c

F2B/F2P=c/a=BI/PI=e