(1)根据椭圆定义,P点轨迹是椭圆,焦点坐标为F1(0,-√3),F2(0,√3),
长半轴在Y轴,
方程为:y^2/4+x^2=1.
(2),设OA向量=（x1,y1),OB向量=（x2,y2),
向量OA⊥OB,
OA·OB=x1*x2+y1*y2=0,
y1=kx1+1,y2=kx2+1,
x1*x2+(kx1+1)(kx2+1)=0,
(x1+x2)(1+k^2)+k(x1+x2)+1=0,(1)
把y=kx+1代入椭圆方程,
(4+k^2)x^2+2kx-3=0,
根据韦达定理,
x1+x2=-2k/(4+k^2),
x1x2=-3/(4+k^2),
代入（1）式,
k^2=1/4,
k=±1/2,
17x^2±4x-12=0,
x1+x2=±4/17,
x1x2=-12/17,
|AB|=√(1+1/4)[(x1+x2)^2-4x1x2]
=(1/2)√5[(4/17)^2+48/17]
=4√65/17,
和你结果一样.漏了（1+k^2).