Question

已知双曲线X^2/a^2-y^2/b^2=1(a,b∈R+)的离心率e∈[√2,2],则一条渐近线与实轴所构成的角的取值范围是？

如上题

Answer 1

渐近线斜率k=b/a
k²=b²/a²=(c²-a²)/a²
=c²/a²-1
=e²-1
√2<=e<2
2<=e²<=4
1<=k²<=3
我们取k>0
则1<=k<=√3
所以夹角范围是[π/4，π/3]