- 圆锥曲线已知AB是抛物线y2=4x的焦点弦,且A(x1,y1),
- 已知AB是抛物线y2=4x的焦点弦,且A(x1,y1),B(x2,y2),点F是抛物线的焦点,过A,B向抛物线的准线L作垂线,垂足分别为A1,B1,直线A1,B1,与x轴交于点K。求证(1)|A1K|,|KF|,|B1K|成等比数列(2)1/|AF| ,1/|KF|,1/|BF|成等差数列
- 抛物线y^2=4x①的焦点为F(1,0),准线L:x=-1.K(-1,0).
设AB:x=my+1,②代入①得
y^-4my-4=0,
则y1+y2=4m,y1y2=-4,
(1)|A1K|=|y1|,|KF|=2,|B1K|=|y2|,
∴|A1K|*|B1K|=|y1y2|=4=|KF|^,
∴|A1K|,|KF|,|B1K|成等比数列.
(2)|AF|=x1+1,|BF|=x2+1,
由②,x1+x2=m(y1+y2)+2=4m^+2,
x1x2=(my1+1)(my2+1)=m^y1y2+m(y1+y2)+1=1,
(x1+1)(x2+1)=x1x2+(x1+x2)+1=4m^+4,
∴1/|AF|+1/|BF|
=(x1+x2+2)/[(x1+1)(x2+1)]
=(4m^+4)/(4m^+4)
=1=2/|KF|,
∴1/|AF| ,1/|KF|,1/|BF|成等差数列.