Let x and y be real numbers such that x^2 + y^2 = 2x + 4y + 2x + 8y. Find the largest possible value of x.
+2668 The largest possible value of x is the x-coordinate of the center + radius
We find both as follows:
Simplify: \(x^2+y^2-12y-4x=0\)
Completing the square, we have: \((x-2)^2+(y-6)^2=40\)
The x-co-ordinate is 2, and the radius is \(\sqrt{40}\)
Thus, the answer is \(\color{brown}\boxed{2+2\sqrt{10}}\)
