To find the radius of the circle from the equation \( x^2 + y^2 - 14x - 4y = 11 \), we will complete the square for both \( x \) and \( y \).
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Rearrange the equation:
\[ x^2 - 14x + y^2 - 4y = 11 \]
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Complete the square for \( x \):
For the term \( x^2 - 14x \):
- Take half of the coefficient of \( x \), which is \(-14\), giving \(-7\).
- Square it: \((-7)^2 = 49\).
So, we rewrite \( x^2 - 14x \) as:
\[ (x - 7)^2 - 49 \]
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Complete the square for \( y \):
For the term \( y^2 - 4y \):
- Take half of the coefficient of \( y \), which is \(-4\), giving \(-2\).
- Square it: \((-2)^2 = 4\).
So, we rewrite \( y^2 - 4y \) as:
\[ (y - 2)^2 - 4 \]
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Substitute back into the equation:
Replacing \( x^2 - 14x \) and \( y^2 - 4y \) in the equation we have:
\[ (x - 7)^2 - 49 + (y - 2)^2 - 4 = 11 \]
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Combine constant terms:
\[ (x - 7)^2 + (y - 2)^2 - 53 = 11 \]
Adding \( 53 \) to both sides gives:
\[ (x - 7)^2 + (y - 2)^2 = 64 \]
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Identify the center and radius:
The equation \((x - 7)^2 + (y - 2)^2 = 64\) is in standard form \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center of the circle and \( r \) is the radius.
Here, \( r^2 = 64 \), so:
\[ r = \sqrt{64} = 8 \]
Thus, the radius of the circle is:
\[ \text{radius} = 8 \]