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avatar+257 

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.

 Jun 25, 2024
 #1
avatar+129725 
+1

Simplify as

 

x^2 -21x + y^2 + 13y  = 25      complete the square on x and  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 = sqrt [ 355/2]  =  sqrt [ 710] / 2

 

The greatest value of x =  21/2 +sqrt (710 ) / 2  =    [ 21 + sqrt 710 ] / 2

 

 

cool cool cool

 Jun 25, 2024

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