y=kx+1代入3x^2-y^2=1
3x^2-k^2x^2-2kx-1=1
(3-k^2)x^2-2kx-2=0
x1+x2=2k/(3-k^2)
x1x2=-2/(3-k^2)
y1y2=(kx1+1)(kx2+1)=k^2x1x2+k(x1+x2)+1=-2k^2/(3-k^2)+2k^2/(3-k^2)+1=1
又OA垂直于OB,则有x1x2+y1y2=0
-2/(3-k^2)+1=0
2=3-k^2
k^2=1
k=(+/-)1
即直线L的倾斜角是45度或135度.