To derive the equation of a circle in standard form, we use the formula:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
where \((h, k)\) is the center of the circle and \(r\) is the radius.
Given:
- Center \((h, k) = (-1, 3)\)
- Radius \(r = 5\)
We can substitute these values into the formula:
-
Substitute \(h\) and \(k\): \[ (x - (-1))^2 + (y - 3)^2 = 5^2 \]
-
Simplify: \[ (x + 1)^2 + (y - 3)^2 = 25 \]
Thus, the equation of the circle in standard form is:
\[ (x + 1)^2 + (y - 3)^2 = 25 \]
From the options you provided, the correct equation is:
\[ (X + 1)^2 + (y - 3)^2 = 25 \]