已知抛物线y的平方=2px的焦点为F,准线交x轴于点C,A,B是抛物线上的两点,若A,B,C三点共线,且2

Question

已知抛物线y的平方=2px的焦点为F,准线交x轴于点C,A,B是抛物线上的两点,若A,B,C三点共线,且2AB=AF+BF,直线AB的斜率为k,则k的平方为

Answer 1

y^2=2px,F(0.5p,0)
C(-0.5p,0)
AB:y=k(x+0.5p)
x=(y-0.5pk)/k
y^2=2px=2p*(y-0.5pk)/k
ky^2-2py+k*p^2=0
xA+xB=2p/k,xA*xB=p^2
yA+yB=k(xA+0.5p)+k(xB+0.5p)=k(xA+xB+p)=k(2p/k+p)=2p+kp
yA*yB=k(xA+0.5p)*k(xB+0.5p)=k^2*[xA*xB+0.5p(xA+xB)+0.25p^2]=k*p^2+1.25k^2*p*2

2AB=2√[(xA-xB)^2+(yA-yB)^2]
=2√[(xA+xB)^2-4xA*xB+(yA+yB)^2-4yA*yB]
=2√[(2p/k)^2-4p^2+(2p+kp)^2-4(kp^2+1.25k^2*p^2]
=2√[(4p^2-4p^2*k^4)/k^2]

AF+BF=√[(xA-0.5p)^2+(yA)^2]+√[(xB-0.5p)^2+(yB)^2]
=√(xA+0.5p)^2+√(xB+0.5p)^2
=xA+xB+p
=2p/k+p=(2p+kp)/k
2AB=AF+BF
2√[(4p^2-4p^2*k^4)/k^2]=(2p+kP)/k
16k^4+k^2+4k-12=0
k=