过准线与x轴交点直线与抛物线交于两点A,B坐标为（x1,y1)(x2,y2)x1y2x2y1怎么求或者求AF,BF斜率和

Question

过准线与x轴交点直线与抛物线交于两点 A,B 坐标为（x1,y1) (x2,y2) x1y2 x2y1怎么求 或者求AF,BF斜率和

Answer 1

解设抛物线方程为
y²=2px,则准线为x=-p/2,准线与x轴的交点为（-p/2,0)
点 A,B 坐标为（x1,y1) ,(x2,y2)
则y1²=2px1,y2=2px2
则x1y2 +x2y1
=x1*2px2+x2*2px1
=2px1x2+2px1x2
=4px1x2
设过准线与x轴交点直线L的直线方程为y=k（x+p/2)
由y=k（x+p/2)
与y²=2px
消y得
[k（x+p/2)]²=2px
即k²x²+pkx+p²k²/4-2px=0
即k²x²+pkx-2px+p²k²/4=0
即x1x2=c/a=（p²k²/4）/k²=p²/4
即x1y2 +x2y1
=x1*2px2+x2*2px1
=2px1x2+2px1x2
=4px1x2
=4p*p²/4
=p³