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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.

 Mar 22, 2024
 #1
avatar+128406 
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x^2 - 4x + y^2 - 2y  = 0          complete the square on x,y

 

x^2 - 4x + 4 + y^2 - 2y + 1 =  4 +1

 

(x- 2)^2 + ( y - 1)^2  =  5

 

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

 

The largest value of x =  2 + sqrt 5

 

 

cool cool cool

 Mar 23, 2024

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