Question

已知P為拋物線(xiàn)y=x²上的動(dòng)點(diǎn),定點(diǎn)A(a,0)關(guān)于P點(diǎn)的對(duì)稱(chēng)點(diǎn)是Q.

(1)求點(diǎn)Q的軌跡方程;

(2)若(1)中的軌跡與拋物線(xiàn)y=x²交于B、C兩點(diǎn),當(dāng)AB⊥AC時(shí),求a的值.

Answer 1

1. y=(x+a)²為所求的軌跡方程.

2.a=±.

解析:

(1)設(shè)P、Q的坐標(biāo)分別為(x₀,y₀)和(x,y),由對(duì)稱(chēng)性得x₀=,y₀=,又y₀=x₀²,

∴=,即y=(x+a)²為所求的軌跡方程.

(2)由得x²-2ax-a²=0,∴∵x₁≠x₂,∴a≠0.

∵AB⊥AC,∴=-1,即y₁y₂+x₁x₂-a(x₁+x₂)+a²=0.

∵y₁·y₂=x₁²·x₂²=(x₁·x₂)²=a⁴,∴a⁴-a²-2a²+a²=0,即得a=±.