To find the equation of a circle, we use the standard form of the equation of 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 the center \((h, k) = (-2, 4)\), we can substitute these values into the equation:
\[ (x + 2)^2 + (y - 4)^2 = r^2 \]
Next, we need to find the radius \(r\). To do this, we calculate the distance between the center \((-2, 4)\) and the point \((3, -4)\) that the circle passes through, using the distance formula:
\[ r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of the center \((-2, 4)\) as \((x_1, y_1)\) and the point \((3, -4)\) as \((x_2, y_2)\):
\[ r = \sqrt{(3 - (-2))^2 + (-4 - 4)^2} \] \[ = \sqrt{(3 + 2)^2 + (-4 - 4)^2} \] \[ = \sqrt{5^2 + (-8)^2} \] \[ = \sqrt{25 + 64} \] \[ = \sqrt{89} \]
Now we have \(r^2 = 89\). We can substitute this back into the equation of the circle:
\[ (x + 2)^2 + (y - 4)^2 = 89 \]
Thus, the equation of the circle is:
\[ \boxed{(x + 2)^2 + (y - 4)^2 = 89} \]
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