Let x and y be real numbers such that x^2 + y^2 = 4(x + y). Find the largest possible value of x.

x^2 + y^2 = 4(x + y).=

x^2 -4x       + y^2-4y      = 0

(x-2)^2        + ( y-2)^2    =  8           this is a circle with center at  2,2    and   r = sqrt 8       max will be    2 + sqrt 8

x² + y²  =  4(x + y)     is an equation of a circle.

 

x² + y²  =  4(x + y)      --->                                   x² + y²  =  4x + 4y

                                             (x² - 4x      ) + (y² - 4y      )  =  0

Complete the squares:        (x² - 4x + 4) + (y² - 4y + 4)  =  8

Factor:                                                 (x - 2)² + (y - 2)²  =  8

 

This circle has its center at  (2, 2)  and has a radius of  sqrt(8).

 

The largest x-value occurs at the right end of the circle  --  the x-value of the center plus the radius  -- 2 + sqrt(8)