intersection of 2 surfaces for line integral

Question

I am asked to find the curve of intersection of the following:

1. $x+y=2$ and
2. $x^2+y^2+z^2=2(x+y)$.

I suppose (1) is a cylinder and unable to find what (2) is, however I think their intersection will be for those (x,y) values on the line $x+y=2$. So that I can parametrize their intersection with respect to x. I want to know whether this will work or, is there any easier or better ways to do the same? Thank you

Answer 1

HINT

(2) transforms using $x^2-2x+1 = (x-1)^2$ into $$(x-1)^2+(y-1)^2 + z^2 = 2$$