Question

已知抛物线y²=4x的准线与双曲线x²/a²-y²/4=1交于A、B两点点F为抛物线的焦点，

若△FAB为正三角形，则双曲线的离心率是？

Answer 1

y²=4x = 2px, p = 2
准线x = -1, F(1, 0)
准线交x轴于C(-1, 0)
CF = 2
tan∠AFC = tan30˚ = AC/CF = AC/2 = √3/3
AC = 2√3/3
A(-1, 2√3/3)
代入x²/a²-y²/4=1, a² = 4/3
c² = a² + b² = 4/3 + 4 = 16/3
e = c/a = (4/√3)/(2/√3) = 2