过点P的直线为:y=k(x-4)+2=kx-4k+2
代入C:(1+k^2)x^2+4(k-2k^2-1)x+(2-4k)^2+1=0
弦坐标为:(x1,y1),(x2,y2), y1=kx1-4k+2, y2=kx2-4k+2
由韦达定理:x1+x2=4(2k^2-k+1)/(1+k^2)
中点坐标为:(x,y)
x=(x1+x2)/2=2(2k^2-k+1)/(1+k^2)
k=(y-2)/(x-4), 代入上式得:
x=2[2(y-2)^2/(x-4)^2-(y-2)/(x-4)+1]/(1+(y-2)^2/(x-4)^2]
即轨迹为:x[(x-4)^2+(y-2)^2]=2[2(y-2)^2-(y-2)(x-4)+(x-4)^2]