Question

求一道抛物线题目!若AB是抛物线y^2=2px的任一焦点弦且A(X1,Y1),B(X2,Y2)则X1X2=?Y1Y2=?

Answer 1

焦点(p/2,0)

若AB斜率不存在

则垂直x轴,是x=p/2

则x1=x2=p/2

x1x2=p^2/4

y^2=2px=p^2

所以y1y2=-p^2

若斜率存在

则y-0=k(x-p/2)

代入抛物线

k^2(x-p/2)^2=2px

k^2x^2-(pk^2+2p)x+k^2p^2/4=0

x1x2=(k^2p^2/4)/k^2=p^2/4

x1+x2=(pk^2+2p)/k^2

y1=k(x1-p/2),y2=k(x2-p/2)

所以y1y2=k^2[x1x2-p/2*(x1+x2)+p^2/4]=k^2*[p^2/4-p/2*(pk^2+2p)/k^2+p^2/4]

=k^2(p^2/2-p^2/2-p^2/k^2)

=-p^2

综上

x1x2=p^2/4

y1y2=-p^2