抛物线y^2=2px(p>0)上任一点Q到顶点O的距离与到焦点F的距离之比是k,求k的取值范围.

题目

抛物线y^2=2px(p>0)上任一点Q到顶点O的距离与到焦点F的距离之比是k,求k的取值范围.

答案

抛物线的参数方程是x=2pt^2 y=2pt抛物线上一点Q(2pt^2,2pt)|OQ|=2pt √(t^2+1)|QF|=2pt^2+1/2pk=(2pt √(t^2+1))/(2pt^2+1/2p)=4t√(t^2+1)/(4t^2+1)=4/√3*(√3t)√(t^2+1)/((4t^2+1))≤2/√3×（3t^2+t^2+1）/(4t^...