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The equation x^2 + y^2 + 10x - 8y = 59 can be written in the form (x - h)^2 + (y - k)^2 = r^2, where h, k, and r are integers. Find the value of r^2.

 Aug 13, 2022
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We have the equation \(x^2 + 10x + y^2 - 8y = 59 \)

 

Add \((10 \div 2)^2 = 5^2 = 25\) to both sides to complete the square on x and \((-8 \div 2)^2 = (-4)^2 = 16\) to both sides to complete the square on y. 

 

This gives us \(x^2 + 10x + 25 + y^2 - 8y + 16 = 59 + 25 + 16\)

 

Completing the square on x and y gives us \((x+5)^2 + (y-4)^2 = 100\), meaning \(r^2 = \color{brown}\boxed{100}\)

 Aug 13, 2022

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