這種題 你就不能怕麻煩,就得死算.
（1） e=c/a = √3,a^2/c =√3/3
a=1,c = √3,b =√2,雙曲線方程為
2x^2 -y^2 = 2
x^2+y^2=2上動點P(x0,y0)(x0y0≠0)處的切線方程為
x0x+y0y = 2
(2)
設(shè)A,B,的坐標為（Xa,Ya),(Xb,Yb),
則（Xa,Ya),(Xb,Yb) 為方程組
x0x+y0y = 2 (1)
2x^2-y^2 = 2 (2)
的解
(1) 代入（2）消去y,得到
(2-x0^2/y0^2)/x^2 +4x0x/y0^2 - (4/y0^2+2) = 0
XaXb = - (4/y0^2+2)/(2-x0^2/y0^2) = -(4+2y0^2)/(2y0^2-x0^2)
(1) 代入（2）消去x,得到
(2y0^2-x0^2)y^2 -8y0y + 8-2x0^2 = 0
YaYb = (8-2x0^2)/(2y0^2-x0^2)
XaXb+YaYb
= -(4+2y0^2)/(2y0^2-x0^2) + (8-2x0^2)/(2y0^2-x0^2)
= [4-2(x0^2+y0^2)]/(2y0^2-x0^2)
(x0,y0) 是圓x^2+y^2=2的點,上式分母為0,
XaXb+YaYb ＝ 0
向量OA和OB垂直,∠AOB = 90度