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Let x and y be real numbers such that x^2 + y^2 = 2x + 4y + 3x + 9y.  Find the largest possible value of x.

 Apr 20, 2022
 #1
avatar+124596 
+1

x^2  +  y^2  = 2x + 4y + 3x + 9y

 

x^2  +  y^2   = 5x + 13y

 

x^2 - 5x  + y^2  - 13x    = 0             complete the square on  x and  y

 

x^2  - 5x + 25/4  + y^2 - 13x + 169 /4    =  25/4 + 169/4

 

(x - 5/2)^2  +  (y - 13/2)^2  =  190 / 4

 

This is a  circle  centered  at  (5/2 , 13/2)  with a radius of  sqrt (190)   / 2

 

x will be maxed when  y =  13/2

 

It's largest possible value =   [5 + sqrt (190) ]   / 2   ≈   9.39

 

 

cool cool cool

 Apr 20, 2022
 #2
avatar+115 
-4

see another correct answer

Kakashi  Apr 20, 2022

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