Question

抛物线的焦点弦交抛物线于A（x1,y1)、B（x2,y2),那么可以得到结论：x1x2=p2/4,y1y2=-p2,如何

Answer 1

当AB垂直于x轴时,方程为x=p/2,代入y^2=2px可得y^2=p^2,得

y1=-p,y2=p,x1=x2=p/2,计算可得.

当AB不垂直与x轴时,设方程为y=k(x-p/2),由y^2=2px得x=y^2/2p代入直线方程化简得ky^2-2py-kp^2=0,所以y1y2=(kp^2)/k=-p^2

x1x2=(y1^2/2p)*(y2^2/2p)=(y1y2)^4/(4p^2)=p^2/4