Untitled 30

1. 设 \(\displaystyle B\) 是椭圆 \(\displaystyle C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \ (a > b > 0)\) 的上顶点，若 \(\displaystyle C\) 上任意一点 \(\displaystyle P\) 都满足 \(\displaystyle |PB| \leqslant 2b\)，则 \(\displaystyle C\) 的离心率的取值范围是 （ ） \choice{\(\displaystyle \left[ \frac{\sqrt{2}}{2}, 1 \right)\)}{\(\displaystyle \left[ \frac{1}{2}, 1 \right)\)}{\(\displaystyle \left( 0, \frac{\sqrt{2}}{2} \right]\)}{\(\displaystyle \left( 0, \frac{1}{2} \right]\)}