已知抛物线有Y^2=2px(p>0)的焦点弦AB的两端点为?
2010-01-14 19:54:38傻***
已知抛物线有 Y^2=2px(p>0)的焦点弦AB的两端点为A(x1已知抛物线有 Y^2=2px(p>0)的焦点弦AB的两端点为A(x1,y1),B(x2,y2),则关已知抛物线有 Y^2=2px(p>0)的焦点弦AB的两端点为A(x1,y1),B(x2,y2),则关系式y1y2/x1x2的值一定等于?已知抛物线有Y^2=2px(p0)的焦点弦AB的两端点为A(x1已知抛物线有Y^2=2px(p0)的焦点弦AB的两端点为A(x1,y1),B(x?
最佳回答
解:弦AB斜率
k=(y1-y2)/(x1-x2)
=(y1-y2)/[(y1^2/2p)-(y2^2/2p)]
=2p/(y1+y2) (1)
而A、F、B三点共线,故
k=(y1-0)/(x1-p/2) (2)
由(1)、(2)得
y1/(x1-p/2)=2p/(y1+y2)
--->y1y2+y1^2=2px1-p^2
而y1^2=2px1
故y1y2=-p^2 (3)
又x1x2=(y1^2/2p)×(y2^2/2p)
=(y1y2)^2/(4p^2)
=(-p^2)^2/(4p^2)
故x1x2=p^2/4 (4)
因此,由(4)÷(3)得
(y1y2)/(x1x2)=-4。
2010-01-14 21:37:19
抛物线焦点弦有以下性质:
x1*x2=p^2/4,y1*y2=-p^2
所以:y1y2/x1x2=-4
证明如下:
LAB:y=k(x-p/2)代入抛物线Y^2=2px(p>0)整理得:
k^2x2-p(k^2+2)x+(kp)^2/4=0
==>x1x1=p^2/4
y1y2=k(x1-p/2)k(x2-p/2)=-p^2
2010-01-14 21:00:12
很赞哦! (90)