If $$ x,y $$ are real numbers such that $$x^2+y^2+2 ≤ 2x+2y $$ , find the value of $$ x+y $$

Question

If $$ x,y $$ are real numbers such that $$x^2+y^2+2 ≤ 2x+2y $$ ,  find the value of $$ x+y $$

Uniteasy Admin · Accepted Answer

$$ x^2+y^2+2 ≤ 2x+2y $$
$$ x^2-2x+1+y^2-2y+1\leq 0 $$
$$ (x-1)^2+(y-1)^2\leq 0 $$
Since addition of two squares is never less than 0, we get the following equation.
$$ (x-1)^2+(y-1)^2= 0 $$
Therefore
$$ x=1 $$ , $$ y=1 $$
Finally
$$ x+y = 2 $$

Question

Answer