P为椭圆上任意一点,F1,F2为椭圆的焦点,设∠PF1F2=45,∠PF2F1=15,则离心率e=

Question

P为椭圆上任意一点,F1,F2为椭圆的焦点,设∠PF1F2=45,∠PF2F1=15,则离心率e=

Answer 1

设椭圆的长轴为2a,焦距为f,则e=f/a
在△PF1F2中,∠P=180°-∠F1-∠F2=180°-45°-15°=120°
根据正弦定理,有F1F2/sinP=PF1/sinF2=PF2/sinF1
由于F1F2=2f,所以2f/sin120°=PF1/sin15°=PF2/sin45°
则PF1=2fsin15°/sin120°,PF2=2fsin45°/sin120°
所以PF1+PF2=2f(sin15°+sin45°)/sin120°
而P在椭圆上,所以PF1+PF2=2a,即2a=2f(sin15°+sin45°)/sin120°
则f/a=sin120°/(sin15°+sin45°)=√3/2/((√6-√2)/4+√2/2)=(3√2-√6)/2
即离心率e=(3√2-√6)/2