已知双曲线x2a2-y2b2=1(a＞0，b＞0)的离心率的取值范围是e∈[233，2]，则两渐近线夹角的取值范围是．

Question

已知双曲线 (x2)/(a2) -(y2)/(b2) =1(a＞0，b＞0) 的离心率的取值范围是 e∈[(23)/(3) ，2] ，则两渐近线夹角的取值范围是______．

Answer 1

∵e= (c)/(a) ， e∈[(23)/(3) ，2] ，
∴ (23)/(3) ≤ (c)/(a) = (a2 + b2)/(a) ≤2
解得 (3)/(3) ≤ (b)/(a) ≤

3

，
设两渐近线构成的角为θ
则渐进线的斜率k=tan (θ)/(2)
∴tan (θ)/(2) = (b)/(a)
即 (3)/(3) ≤tan (θ)/(2) =≤

3

，
∴ (π)/(6) ≤ (θ)/(2) ≤ (π)/(3)
∴ (π)/(3) ≤θ≤ (2π)/(3)
∴两渐近线夹角的取值范围是[ (π)/(3) ， (2π)/(3) ]
故答案为[ (π)/(3) ， (2π)/(3) ]