Question

抛物线y的平方=4x的准线与双曲线x²／a²-y²/4=1交与A,B两点，f是抛物线焦点，fab是正三角形

抛物线y的平方=4x的准线与双曲线x²／a²-y²/4=1交与A,B两点，f是抛物线焦点，fab是正三角形，求离心率

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

追问

答案是根号51比3.而且我算a² = 3比4、?

追答

A(-1, 2√3/3)
代入x²/a²-y²/4=1, 1/a²-（4/3)/4=1，1/a²=4/3，a² =3/4
c² = a² + b² = 3/4 + 4 = 19/4
e = c/a = (√19)/(√3) =根号57/3