已知A(8,0),B,C两点分别在y轴上和x轴上运动,并满足向量AB*BP=0向量BC=CP向量,求P动点的轨迹

Question

已知A(8,0),B,C两点分别在y轴上和x轴上运动,并满足向量AB*BP=0向量BC=CP向量,求P动点的轨迹

Answer 1

1.设B(0,b),C(c,0),P(x,y),则
向量AB=(-8,b),BP=(x,y-b),
∴向量AB*BP=-8x+b(y-b)=0,①
向量BC=CP得(c,-b)=（x-c,y),
∴b=-y,
代入①,化简得y^2=4x,为所求.
2.设L:x=my+8,②代入上式,整理得
y^2-4my-32=0,
设M(x1,y1),N(x2,y2),则
y1+y2=4m,y1y2=-32.
由②,x1+x2=m(y1+y2)+16=4m^2+16,
x1x2=(my1+8)(my2+8)=m^2*y1y2+8m(y1+y2)+64=64.
∴向量QM*向量QN=(x1+1)(x2+1)+y1y2
=x1x2+(x1+x2)+1+y1y2
=64+4m^2+16+1-32
=4m^2+49=97,
解得m=土2√3,
∴L:x土2y√3-8=0.