F1.F2是定点P是以F1.F2为公共焦点的椭圆和双曲线交点,F1垂直F2,e1.e2是椭圆.双曲线离心率

F1P垂直F2P
设椭圆的方程
x^2/a^2+y^2/b^2=1
双曲线的方程
x^2/m^2-y^2/n^2=1
F1P+F2P=2a F1P^2+2F1PF2P+F2P^2=4a^2 (1)
F1P-F2P=2m F1P^2-2F1PF2P+F2P^2=4m^2 (2)
F1P^2+F2P^2=4c^2
(1)+(2)得
F1P^2+F2P^2=2a^2+2m^2=4c^2
即a^2+m^2=2c^2
a^2/c^2+m^2/c^2=2
1/e1^2+1/e2^2 =a^2/c^2+m^2/c^2=2