已知F1,F2是椭圆的两个焦点,P是椭圆上一点,若∠PF1F2=15,∠PF2F1=75,则椭圆的离心率为?

已知F1,F2是椭圆的两个焦点,P是椭圆上一点,若∠PF1F2=15,∠PF2F1=75,则椭圆的离心率为?

F1F2=2c
a^2=b^2+c^2
PF2=2csin(15),PF1=2csin(75)
PF1+PF2=2a=2c(sin(15)+sin(75))=
=2c(sin(45-30)+sin(45+30))=2c*2sin(45)cos(30)=c*sqrt(6)
a=c*sqrt(6)/2
a^2=c^2*6/4=3/2*c^2=c^2+b^2
b^2=c^2/2
b=csqrt(2)
离心率=(a-b)/(a+b)=(sqrt(6)-sqrt(2))/(sqrt(6)+sqrt(2))=(sqrt(6)-sqrt(2))^2/4