点P是椭圆x2/a2+y2/b2=1上一点,F1F2是焦点,且∠PF1F2=105,∠PF2F1=15,求e.

点P是椭圆x2/a2+y2/b2=1上一点,F1F2是焦点,且∠PF1F2=105,∠PF2F1=15,求e.
点P是椭圆x2/a2+y2/b2=1上一点,F1F2是焦点,且∠PF1F2=105,∠PF2F1=15,求e.

点P是椭圆x2/a2+y2/b2=1上一点,F1F2是焦点,且∠PF1F2=105,∠PF2F1=15,求e.
sin15°=sin(45°-30°)=（根号6-根号2）/4
cos15°=sin75°=sin105°=cos(45°-30°)=（根号6+根号2）/4
△PF1F2中
∠PF1F2=105°,∠PF2F1=15°
∠P=60°
利用正弦定理求出PF1=F1F2*sin15°/sin60°=(3根号2+根号6)c/3
PF2=F1F2*sin105°/sin60°=(3根号2-根号6)c/3
PF1+PF2=2根号2c=2a
e=c/a=根号2/2