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Let $x$ and $y$ be real numbers.  If $x$ and $y$ satisfy
x^2 + y^2 = 4x + 2y
then find the largest possible value of $x.$ Give your answer in exact form using radicals, simplified as far as possible.

 Jan 2, 2024
 #1
avatar+126665 
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x^2 - 4x + y^2 - 2y  = 0

 

x^2 - 4x + 4  + y^2 - 2y + 1  =  5

 

(x - 2)^2  + (y - 1)^2  =  5

 

 

This is a  circle  centered at  (2, 1)  with a radius of sqrt (5)

 

x max  =  2 + sqrt (5)

 

 

cool cool cool

 Jan 3, 2024

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