Question

F为椭圆左焦点,A,B为椭圆的右顶点和上顶点,P为椭圆上的点,且PO//AB,PF⊥FA,椭圆离心率是多少?

Answer 1

假设椭圆：x^2/a^2+y^2/b^2=1,F(-c,0),A(a,0),B(0,b)

PF垂直于FA,P(x,y)：x= -c,y=±b^2/a

PO//AB,k(PO)=y/x=k(AB)= -b/a,

所以,y=b^2/a

y/x= -b/a= -b^2/ac得到：b=c

所以,a^2=b^2+c^2=2c^2

e=c/2=√2/2