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Let x and y be real numbers such that x^2 + y^2 = 6(x+y). Find the largest possible value of y.

 Dec 16, 2020
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x^2  + y^2  = 6(x + y)

 

x^2 + y^2  = 6x + 6y

 

x^2   - 6x  + y^2 - 6y   = 0       complete the  square  on x,y

 

x^2  -6x + 9  + y^2  - 6y + 9   =  9 + (

 

(x -3)^2  + (y - 3)^2   = 18

 

This is a circle with a center of  (3,3)   and a  radius of  sqrt (18)

 

The max value of  y   is     3 + sqrt (18)    =    3 + 3sqrt (2)

 

cool cool cool

 Dec 16, 2020

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