Question

設(shè)A(x₁,y₁)、B(x₂,y₂)兩點(diǎn)在拋物線y=2x²上,l是AB的垂直平分線.

(1)當(dāng)且僅當(dāng)x₁+x₂取何值時(shí),直線l經(jīng)過拋物線的焦點(diǎn)F?證明你的結(jié)論.

(2)當(dāng)直線l的斜率為2時(shí),求l在y軸上的截距的取值范圍.

Answer 1

l在y軸上截距的取值范圍是(,+∞).

解析:

(1)F∈l|FA|=|FB|A、B兩點(diǎn)到拋物線準(zhǔn)線的距離相等y₁=y₂x₁²=x₂²(x₁+x₂)

(x₁-x₂)=0,由A、B為兩點(diǎn),∴x₁≠x₂x₁+x₂=0.

故x₁+x₂=0時(shí),直線l經(jīng)過焦點(diǎn)F.

(2)設(shè)l:y=2x+b,

∵l⊥AB,

設(shè)AB：y=-x+m,聯(lián)立y=2x²,

∴2x²+x-m=0,

設(shè)AB中點(diǎn)N(x₀,y₀),

則

由點(diǎn)N∈l,∴+m=-+b,

b=+m>-=.

∴l(xiāng)在y軸上截距的取值范圍是(,+∞).