Question

给定抛物线C :y2 =2px(p0 ) ,O是顶点 ,过y轴上一定点M(0 ,m) (m ≠ 0 )引直线交C于P、Q

Answer 1

设直线为 y=kx+m 代入 y^2 =2px(p>0 )中,得 (kx^2) - 2(p-km)x+m^2=0, 设P(x1,y1), Q(x2,y2),

则 x1+x2= 2(p-km)/k^2, x1x2=m^2/k^2, y1=kx1+m , y2=kx2+m ,

KOP+KOQ =y1/x1+y2/x2=(kx1+m)/x1+(kx2+m)/x2

=2k+m(x1+x2)/(x1x2)

=2k+2m(p-km)/m^2

=2p/m.（定值）