设,双曲线C的方程为:Y^2/2-X^2/b^2=1,
则渐近线方程为Y=±√2/b*x,
与圆M:(X-2)²+Y²=1相切,
(x-2)^2+(±√2/b*x)^2=1,
(2/b^2+1)x^2-4x+3=0,
(-4)^2-4(2/b^2+1)*3=0,
b^2=6,
则,双曲线C的方程为:
Y^2/2-X^2/6=1.
设,点P的坐标为(m,n),
d=√2=|m-n-√2|/√(1+1),
2=|m-n-√2|,
而,点P(m,n)在Y^2/2-X^2/6=1,上
有n^2/2-m^2/6=1,
3n^2-m^2-6=0.