C上切线斜率为y'=2ax
设A为(x1,y1),B为(x2,y2)
线AP斜率为(y1+1)/(x1-1)=2ax1
ax1^2+1=y1+1=2ax1(x1-1)
ax1^2-2ax1-1=0.
同理
ax2^2-2ax2-1=0.
那么x1+x2=2a,x1x2=-1
AB斜率为
(y1-y1)/(x1-x2)=(ax1^2-ax2^2)/(x1-x2)=
a(x1-x2)(x1+x2)/(x1-x2)=a(x1+x2)=2a^2.
故AB斜率为定值