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Find the largest real number x for which there exists a real number y such that x^2+y^2 = 2x+2y .

I think it is a circle, but how am i supposed to figure this out??

The standard equation for a circle is ...

(x-a)^2 + (y-b)^2=r^2

x^2+y^2-2x-2y=0

x^2-2x + 1 + y^2-2y+1= 2 That wont work. We do have perfect squares on the left, but there is needed a zero on the right. Hmmm It is not a circle.

Lets look at if y is y<=0
if y is zero, then
x^2=2x or x can be any real number up to infinity, and there can be no larger x. So the answer is x=inf is the largest real number. Yes, y=0 is a real number.

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