题目

题型：不详难度：来源：

直线l在双曲线=1上截得弦长为4,其斜率为2,则直线l在y轴上的截距是_____________.

答案

±

解析

设直线l的方程为y="2x+m.                                                " ①
将①代入双曲线方程,得10x²+12mx+3(m²+2)=0,设l与双曲线的交点为A(x₁,y₁), B(x₂,y₂),由韦达定理可得x₁+x₂=-m,                                                      ②
x₁x₂=(m²+2),                                                               ③
又y₁=2x₁+m,y₂=2x₂+m,
∴y₁-y₂=2(x₁-x₂).再利用②③,∴|AB|²=(x₁-x₂)²+(y₁-y₂)²=5(x₁-x₂)²=5［(x₁+x₂)²-4x₁x₂］=5［m²-4×(m²+2)］.
∵|AB|=4,∴5［m²-(m²+2)］=4².∴3m²=70.∴m=±.