To write the standard form of the equation of a circle, you need the center of the circle and the radius.
Based on the provided information, the center of the circle is at the point \((3, -2)\). The standard form of the equation of a circle 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.
Next, we need to find the radius \(r\). The radius is the distance from the center of the circle to any point on the circle. You have mentioned that the circle passes through multiple points, and we can use any of these points to determine the radius.
Let's use the point \((3, 1)\) to find the radius:
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Calculate the distance from the center \((3, -2)\) to the point \((3, 1)\):
\[ r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(3 - 3)^2 + (1 - (-2))^2} = \sqrt{0 + (1 + 2)^2} = \sqrt{3^2} = 3 \]
So, the radius \(r = 3\), and therefore, \(r^2 = 9\).
Now we can substitute \(h\), \(k\), and \(r^2\) into the standard equation:
\[ (x - 3)^2 + (y + 2)^2 = 9 \]
Thus, the correct standard form of the equation of the circle is:
\[ \left( x - 3 \right)^{2} + \left( y + 2 \right)^{2} = 9 \]
Based on the options provided, the correct answer is:
left-parenthesis x minus 3 right-parenthesis squared plus left-parenthesis y plus 2 right-parenthesis squared equals 9.