Question

15.已知A（2,0）,B（4,0）,动点P在抛物线y*2=-4x上运动,求向量AP*向量BP取得最小值时的点P的坐标

Answer 1

设P点坐标为(x,y),则y²=-4x

则AP=(2-x,-y),BP=(4-x,-y)

∴AP*BP=(2-x)(4-x)+(-y)(-y)

=x²-6x+8+y²

=x²-6x+8-4x

=x²-10x+8

=(x-5)²-17

∵x∈(-∞,0]

则当x=0时,(x-5)²-17取最小值

∴P点坐标为(0,0)