Question

抛物线c的焦点在原点,顶点在x轴的负半轴,动直线:x+y+m=0与x交与A,B两点,当△AOB面积

Answer 1

由已知可设抛物线c的方程为：y^2=2p(x+p/2),与直线：x+y+m=0联立

消x得y^2+2py+2pm-p^2=0设A(x1,y1)、B(x2,y2)

|AB|=sqr(2)*|y1-y1|=sqr(2)*sqr((y1+y2)^2-4y1*y2)=4sqr(p^2-pm)

O点到直线:x+y+m=0的距离为d=m/sqr(2)

则△AOB面积=d*|AB|/2=sqr(m*m*(2p-2m)*p)