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Let x and y be real numbers such that x^2 + y^2 = 2x + 4y + 2x + 8y.  Find the largest possible value of x.

 Apr 19, 2022
 #1
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The largest possible value of x is the x-coordinate of the center + radius

 

We find both as follows: 

 

Simplify: \(x^2+y^2-12y-4x=0\)

 

Completing the square, we have: \((x-2)^2+(y-6)^2=40\)

 

The x-co-ordinate is 2, and the radius is \(\sqrt{40}\)

 

Thus, the answer is \(\color{brown}\boxed{2+2\sqrt{10}}\)

 Apr 19, 2022

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