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(guò)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(,+∞).

(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(guò)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(,+∞).