已知直线y=ax+1与双曲线3x^2-y^2=1相交于A,B两点.

已知直线y=ax+1与双曲线3x^2-y^2=1交于A,B两点
(1)求a的取值范围
(2)若以AB为直径的圆过坐标原点,求实数a的值
(3)若S△OAB=2,求a的值
(4)是否存在这样的实数a,使A,B两点关于直线y=x/2对称?若存在,请求出a的值,若不存在,说明理由.
将y=ax+1代入3x^2-y^2=1,得：(3-a^2)x^2 -2ax -2 =0
1.交于A,B两点 ==> 判别式 =(2a)^2 -4*(3-a^2)(-2)> 0
==> -根号6 < a < 根号6
2.以AB为直径的圆过坐标原点
OA垂直OB,(y1/x1)(y2/x2) = -1,x1x2+y1y2=0
x1x2+(a*x1 +1)(a*x2 +1) =0
(1+a^2)x1x2 +a(x1+x2)+1 =0 ...(1)
x1x2 =-2/(3-a^2),x1+x2 =2a/(3-a^2)代入(1),得:
a = 1,-1
3.2 = S△OAB =AB*d/2,d =1/根号(1+a^2) = 点O到直线AB距离
AB = |x1-x2|*根号(1+a^2)
==> |x1-x2| =4,(x1+x2)^2 -4x1x2 =16 ...(2)
x1x2 =-2/(3-a^2),x1+x2 =2a/(3-a^2)代入(2),解得:
a = 根号2,-根号2,(根号15)/2,-(根号15)/2
4.A,B两点关于直线y=x/2对称
A,B的中点在直线y=x/2上:
(y1+y2)/2 =[(x1+x1)/2]/2,a(x1+x2)+2 =(x1+x2)/2
x1+x2 =2a/(3-a^2)代入,解得:a=6 > -根号6
因此,不存在实数a,A,B两点关于直线y=x/2对称