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)