To find the radius of the circle represented by the equation \(x^2 + y^2 - 16x - 10y + 40 = 0\), we first need to rewrite the equation in standard form, which is given by \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center of the circle and \(r\) is the radius.
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Rewrite the equation: \[ x^2 - 16x + y^2 - 10y + 40 = 0 \]
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Group and complete the square for \(x\) and \(y\):
- For \(x^2 - 16x\): \[ x^2 - 16x = (x - 8)^2 - 64 \]
- For \(y^2 - 10y\): \[ y^2 - 10y = (y - 5)^2 - 25 \]
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Substitute back into the equation: \[ (x - 8)^2 - 64 + (y - 5)^2 - 25 + 40 = 0 \] Simplifying this: \[ (x - 8)^2 + (y - 5)^2 - 49 = 0 \] Thus, \[ (x - 8)^2 + (y - 5)^2 = 49 \]
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Identify the radius:
- The standard form is \((x - h)^2 + (y - k)^2 = r^2\). Here, \(r^2 = 49\).
- Taking the square root gives \(r = \sqrt{49} = 7\).
Therefore, the radius of the circle is 7.