Question

过椭圆x2/a2+Y2/B2=1的左焦点F1作x轴的垂线交椭圆与P,F2为右焦点,若角F1PF2=60° 则椭圆的离心率

Answer 1

e=根号下3/3.

Answer 2

左焦点坐标是(-c,0)，则点P点坐标是(-c,y)，求出y
因为点P在椭圆上，则
c^2/a^2+y^2/b^2=1
b^2c^2+a^2y^2=a^2b^2
b^2(a^2-b^2)+a^2y^2=a^2b^2
a^2b^2-b^4+a^2y^2=a^2b^2
y^2=b^4/a^2
y=±b^2/a
所以点P点坐标是(-c,±b^2/a)
F1P=|yp|=b^2/a
F2P=2a-F1P=2a-b^2/a=(2a^2-b^2)/a
F1F2=2c

cos∠F1PF2=(F1P^2+F2P^2-F1F2^2)/(2F1PF2P)=1/2
F1P^2+F2P^2-F1F2^2=F1PF2P
b^4/a^2+(2a^2-b^2)^2/a^2-4c^2=b^2/a*(2a^2-b^2)/a
b^4+4a^4-4a^2b^2+b^4-4a^2c^2=2a^2b^2-b^4
3b^4+4a^2-6a^2b^2-4a^2(a^2-b^2)=0
3b^4+4a^2-6a^2b^2-4a^4+4a^2b^2=0
3b^4-2a^2b^2=0
3b^2=2a^2
3(a^2-c^2)=2a^2
3a^2-3c^2=2a^2
a^2=3c^2
a=√3c
e=c/a=√3/3

Answer 3

2/根号2