Question

直線l在雙曲線=1上截得弦長為4,其斜率為2,則直線l在y軸上的截距是_____________.

Answer 1

±

設(shè)直線l的方程為y="2x+m.                                                " ①
將①代入雙曲線方程,得10x²+12mx+3(m²+2)=0,設(shè)l與雙曲線的交點為A(x₁,y₁), B(x₂,y₂),由韋達定理可得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=±.