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avatar+952 

Let $x$ and $y$ be real numbers. If $x$ and $y$ satisfy
x^2 + y^2 = 4x - 8y + 17x - 5y + 25,
then find the largest possible value of $x.$ Give your answer in exact form using radicals, simplified as far as possible.

 Apr 4, 2024
 #1
avatar+128794 
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Simplify  as

 

x^2 - 21x + y^2 + 13y  =  25     complate the square on x, y

 

x^2 -21x + 441/4   + y ^2 + 13y + 169/4 =   25 + 441/4 + 169/4

 

(x - 21/2)^2  + (  y +13/2)^2 =  355/2

 

This is a circle centered at  ( 21/2 , - 13/2)  with a radius of sqrt [ 355 / 2 ] 

 

Largest value of x =  21/2  + sqrt [ 355 / 2 ]  =   21/2 + sqrt [ 710 ]  / 2

 

cool cool cool

 Apr 5, 2024

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