设经过点P(0,2)的直线l方程为 y=kx+2
将x=(y-2)/k代入抛物线y^2=4x,得y^2=4(y-2)/k,
即ky^2-4y+8=0,则y1+y2=4/k,y1y2=8/k
设直线与抛物线交点为E(y1^2/4,y1),F(y2^2/4,y2)
已知圆的圆心为C(1,0),则
向量CE=(y1^2/4-1,y1),向量CF=(y2^2/4-1,y2)
m=向量CE*向量CF=(y1^2/4-1)(y2^2/4-1)+(y1y2)
=(y1y2/4)^2-(y1^2+y2^2)/4+1+y1y2
=(y1y2/4)^2-[(y1+y2)^2-4y1y2]/4+1+y1y2
=1/k^2-[16/k^2-32/k]/4+1+8/k
=-3/k^2+16/k+1
=(k^2+16k-3)/k^2
若m>0,则k^2+16k-3>0
解得 k>-8+√67或k