Let x and y be real numbers such that x^2 + y^2 = 2x + 4y + 3x + 9y. Find the largest possible value of x.
+128408 x^2 + y^2 = 2x + 4y + 3x + 9y
x^2 + y^2 = 5x + 13y
x^2 - 5x + y^2 - 13x = 0 complete the square on x and y
x^2 - 5x + 25/4 + y^2 - 13x + 169 /4 = 25/4 + 169/4
(x - 5/2)^2 + (y - 13/2)^2 = 190 / 4
This is a circle centered at (5/2 , 13/2) with a radius of sqrt (190) / 2
x will be maxed when y = 13/2
It's largest possible value = [5 + sqrt (190) ] / 2 ≈ 9.39
