已知椭圆(C):(frac{x^2}{a^2}+frac{y^2}{b^2}=1(a>b>0))的左顶点和上顶点分别为(A),(B),左、右焦点分别是(F_1),(F_2),在线段(AB)上有且只有一个点(P)满足(PF_1ot PF_2),则椭圆的离心率的平方为( )
(A.frac{sqrt{3}}{2})
(B.frac{3-sqrt{5}}{2})
(C.frac{-1+sqrt{5}}{2})
(D.frac{sqrt{3}-1}{2})