Substituting into the curve equation, k (x-1)+1= x 2+1,

Simplified as x 2-kx+k = 0,

Because the straight line is tangent to the curve, the above equation has two roots.

The discriminant is 0,

That is, (-k) 2-4k = 0,

The solution is k=0 or k=4,

So the tangent equation is y= 1 or y=4x-3.