+178 Let $x$ and $y$ be real numbers such that $x^2 + y^2 = 4x + 12y.$ Find the largest possible value of $x + y.$ Give your answer in exact form using radicals, simplified as far as possible.
+1314
x2 + y2 = 4x + 12y is the equation of a circle.
My approach will be to complete the squares
of x and y, to obtain the radius of that circle.
x2 + y2 = 4x + 12y
(x2 – 4x) + (y2 – 12y) = 0
(x2 – 4x + 4) + (y2 – 12y + 36) = 4 + 36
(x – 2)2 + (y – 6)2 = 40
This draws a circle centered at (2, 6) with a radius sqrt(40).
Never mind the center, all that interests us is the radius.
By formula, the maximum x+y is the radius times sqrt(2).
In this problem, the largest x+y is sqrt(40) times sqrt(2).
sqrt(40) • sqrt(2) = sqrt(80) = 4 • sqrt(5)
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