Question

数学题求解：已知抛物线y=x2上两点A、B,且直线AB过抛物线y=x2的焦点F,过A、B分别作抛物线

Answer 1

(1)抛物线y=x^2①的焦点F是（0,1/4）,y'=2x,

设AB:y=kx+1/4,代入①,

x^-kx-1/4=0,

设A(x1,x1^),B(x2,x2^),P(x,y),x1≠x2,则

x1+x2=k,x1x2=-1/4,

PA:y-x1^=2x1(x-x1),y=x1(2x-x1),

PB:y-x2^=2x2(x-x2).y=x2(2x-x2),

解得x=(x1+x2)/2=k/2,

y=x1x2=-1/4.

∴P点轨迹方程是y=-1/4.

(2)PF的斜率=（1/2）/（-k/2)=-1/k,

∴PF⊥AB.