To find the standard form of the equation of the circle, we use the general formula for a circle:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
where \((h, k)\) is the center of the circle and \(r\) is the radius.
Given:
- Center of the circle \( (h, k) = (-2, 2) \)
- Points on the circle \( (1, 0) \) and \( (-5, 0) \)
First, we find the radius \(r\). We can calculate the distance between the center and one of the points on the circle, for example, from \((-2, 2)\) to \((1, 0)\):
\[ r = \sqrt{(1 - (-2))^2 + (0 - 2)^2} = \sqrt{(1 + 2)^2 + (0 - 2)^2} = \sqrt{3^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13} \]
Now that we have the center \((-2, 2)\) and the radius \(r = \sqrt{13}\), we can substitute these values into the circle equation:
\[ (x + 2)^2 + (y - 2)^2 = (\sqrt{13})^2 \]
This simplifies to:
\[ (x + 2)^2 + (y - 2)^2 = 13 \]
Thus, the standard form of the equation of the circle is:
\[ \boxed{(x + 2)^2 + (y - 2)^2 = 13} \]
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