代入
(a^2+1)x^2+2abx+b^2-1=0
x1+x2=-2ab/(a^2+1),x1x2=(b^2-1)/(a^2+1)
y=ax+b
y1y2=(ax1+b)(ax2+b)=a^2x1x2+ab(x1+x2)+b^2
=a^2(b^2-1)/(a^2+1)-2a^2b^2/(a^2+1)+b^2
OA垂直OB
OAOB斜率分别是y1/x1,y2/x2
所以(y1/x1)(y2/x2)=-1
y1y2=-x1x2
所以
a^2(b^2-1)/(a^2+1)-2a^2b^2/(a^2+1)+b^2=-(b^2-1)/(a^2+1)
两边乘a^2+1
a^2(b^2-1)-2a^2b^2+b^2(a^2+1)=-(b^2-1)
a^2b^2-a^2-2a^2b^2+a^2b^2+b^2=-b^2+1
-a^2+b^2=-b^2+1
a^2-2b^2+1=0