The degree measure of the angle between 

$x^2+y^2-2x-2y+1=0$ and line $y=\lambda x+1-\lambda ,$ is

 The angle between a line and a circle is defined as 

the angle made by the portion of the line intercepted 

by the circle at any point on the circle. Here, the line 

passes through the centre of the circle i.e. it is a diameter.

 So, required angle is a right angle.