analytic geometry helps are appreciated
posted by Knights .
For some positive real number r , the line x+y=r is tangent to the circle x^2+y^2 = r. Find r.
How do we do this? Set equal equations together??

r^1/2 is the radius of the circle.
r is the yintercept of the line
y = r  x.
It is tangent to the circle if
r^2 = 2*(sqrtr)^2
(from Pythagorean rule)
Therefore r^2 = 2r
r = 2
Plot the two functions
y = 2x and
x^2 + y^2 = 2
You will see that there is tangency at
(sqrt2, sqrt2) 
thanks a bunch sire

Yes set the two equations equal to each other. You then get that r = 2. I'm not going to tell you all of the work because drwls has already answered.