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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.

 Jun 19, 2024
 #1
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Simplify as

 

x^2 -4x + y^2 - 12y  =  0       complete the square on x,y

 

x^2 -4x + 4 + y^2 - 12y + 36  = 4 + 36

 

(x -2)^2 + (y -6)^2  = 40

 

This is a circle centered at (2,6)  with a radius of sqrt (40)

 

max x + y  can be found by the intersection of the line  y =(x -2) + 6  → y = x + 4  with the circle

 

The intersection point is

 

x =  2 + sqrt 40 * cos 45°      y =  6 + sqrt 40 * sin 45°  

 

max [ x ] + [ y ]  =  [  2 + sqrt (40) / sqrt (2)]  + [ 6 + sqrt (40)/sqrt (2)]   =

 

8 + 2sqrt (20)  =

 

8 + 4sqrt (5)

 

cool cool cool

 Jun 19, 2024
edited by CPhill  Jun 20, 2024

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