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The equation x^2 + 6x + y^2 + 6y = 18 + 2x - 10y represents a circle. What is its center? Write your answer as an ordered pair.

 May 8, 2022
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Whenever you see a circle in the form \(Ax^2 + Ay^2 + Bx + Cy + D = 0\), complete the squares to get something like \((x - h)^2 + (y - k)^2 = r^2\), then the center will just be (h, k) and the radius will just be r.

 

So, for this one, you move every term to the left-hand side first.

\(x^2 + y^2 + 4x + 16y - 18 = 0\)

 

Now, complete the square.

\((x^2 + 4x + 4) - 4 + (y^2 + 16y + 64) - 64 - 18 = 0\\ (x + 2)^2 - 4 + (y + 8)^2 - 64 - 18 = 0\\ (x + 2)^2 + (y + 8)^2 = 86\)

 

Can you take it from here to identify h and k?

 

Hint: You can write \(x + a\) as \(x - (-a)\) at all times.

 May 8, 2022

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