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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. 

 Jan 14, 2024

Best Answer 

 #1
avatar+36919 
+1

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: 

 Jan 14, 2024
 #1
avatar+36919 
+1
Best Answer

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: 

ElectricPavlov Jan 14, 2024
 #2
avatar+128732 
0

Nice, EP   !!!!

 

 

cool cool cool

CPhill  Jan 14, 2024
 #3
avatar+36919 
0

Thanx, Chris !    

~EP

 

 

.....as an edit...the question asked for radicals in the answer 

                 x = 2 + 2 sqrt(5)     y = 6 + 2 sqrt(5) 

ElectricPavlov  Jan 14, 2024

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