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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.

 Mar 25, 2024
 #1
avatar+128399 
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x^2 - 21x + y^2 + 13y   =  25             complete 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)

 

cool cool cool

 Mar 26, 2024

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