+1678 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.
+36943 Find max of x+y start with standard circle form
x^2 -4x + y^2 - 12y = 0
(x-2)^2 + (y-6)^2 = 4 + 36
(x-2)^2 + (y-6)^2 = 40
Now find the 45 degree line that goes through the center (2,6)
slope = 1 y = x + b shows b = 4
so the line is y = x + 4
where the circle and the line intersect will be the max x+y
x+4 = y
sub into the circle equation and solve for x = 6.47124 then y = 10.47124
Mx value of (x+y) is 16.94428
Here is a graph:
+36943 Find max of x+y start with standard circle form
x^2 -4x + y^2 - 12y = 0
(x-2)^2 + (y-6)^2 = 4 + 36
(x-2)^2 + (y-6)^2 = 40
Now find the 45 degree line that goes through the center (2,6)
slope = 1 y = x + b shows b = 4
so the line is y = x + 4
where the circle and the line intersect will be the max x+y
x+4 = y
sub into the circle equation and solve for x = 6.47124 then y = 10.47124
Mx value of (x+y) is 16.94428
Here is a graph:
+36943 Thanx, Chris !
~EP
.....as an edit...the question asked for radicals in the answer
x = 2 + 2 sqrt(5) y = 6 + 2 sqrt(5)
