To find the center and radius of the circle given by the equation \(x^2 + y^2 + 12x - 18y + 17 = 0\), we will complete the square.
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Rearrange the equation: \[ x^2 + 12x + y^2 - 18y + 17 = 0 \] \[ x^2 + 12x + y^2 - 18y = -17 \]
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Complete the square for \(x\):
- Take the coefficient of \(x\) (which is 12), halve it (giving 6), and square it (giving 36).
- Thus, \(x^2 + 12x\) can be rewritten as: \[ (x + 6)^2 - 36 \]
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Complete the square for \(y\):
- Take the coefficient of \(y\) (which is -18), halve it (giving -9), and square it (giving 81).
- Thus, \(y^2 - 18y\) can be rewritten as: \[ (y - 9)^2 - 81 \]
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Substituting back: \[ (x + 6)^2 - 36 + (y - 9)^2 - 81 = -17 \] \[ (x + 6)^2 + (y - 9)^2 - 117 = -17 \] \[ (x + 6)^2 + (y - 9)^2 = 100 \]
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Identify the center and radius: The equation \((x + 6)^2 + (y - 9)^2 = 100\) is in the standard form of a circle \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.
- Center \((h, k) = (-6, 9)\)
- Radius \(r = \sqrt{100} = 10\)
Thus, the center of the circle is \((-6, 9)\) and the radius is \(10\).
The correct answer is: center=(-6,9), radius=10