To find the center of the circle by completing the square, we start with the given equation:
\[ x^2 + y^2 - 10x + 4y = 20 \]
We'll rearrange and isolate terms involving \(x\) and \(y\):
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Rearranging the equation gives: \[ x^2 - 10x + y^2 + 4y = 20 \]
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Next, we'll complete the square for the \(x\) and \(y\) terms.
Completing the square for \(x\):
- Take the coefficient of \(x\), which is -10.
- Divide it by 2: \(-10/2 = -5\).
- Square it: \((-5)^2 = 25\).
- Add and subtract this square inside the equation:
\[ x^2 - 10x + 25 \]
This becomes:
\[ (x - 5)^2 \]
Completing the square for \(y\):
- Take the coefficient of \(y\), which is 4.
- Divide it by 2: \(4/2 = 2\).
- Square it: \(2^2 = 4\).
- Add and subtract this square inside the equation:
\[ y^2 + 4y + 4 \]
This becomes:
\[ (y + 2)^2 \]
- Now, add these squares to the original equation:
Starting from:
\[ (x - 5)^2 + (y + 2)^2 = 20 + 25 + 4 \]
- Combine the constants on the right side:
\[ (x - 5)^2 + (y + 2)^2 = 49 \]
Now, we can see that this is the standard form of the equation of a circle:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
where \((h, k)\) is the center and \(r\) is the radius.
- From \( (x - 5)^2 + (y + 2)^2 = 49\), we find:
- The center \((h, k)\) is \((5, -2)\).
Thus, the center of the circle is:
\[ \boxed{(5, -2)} \]
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