To find the equation of a circle with a given center and radius, we use the standard form equation of a circle:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
Here, [tex]\((h, k)\)[/tex] represents the coordinates of the center of the circle, and [tex]\(r\)[/tex] represents the radius.
Given:
- The center of the circle is [tex]\((-5, 2)\)[/tex], so [tex]\(h = -5\)[/tex] and [tex]\(k = 2\)[/tex].
- The radius of the circle is [tex]\(7\)[/tex], so [tex]\(r = 7\)[/tex].
Substitute the values of [tex]\(h\)[/tex], [tex]\(k\)[/tex], and [tex]\(r\)[/tex] into the standard form equation:
[tex]\[
(x - (-5))^2 + (y - 2)^2 = 7^2
\][/tex]
Simplify the equation:
[tex]\[
(x + 5)^2 + (y - 2)^2 = 49
\][/tex]
Thus, the correct equation of the circle is:
[tex]\[
(x + 5)^2 + (y - 2)^2 = 49
\][/tex]
Among the given options, the correct choice is:
[tex]\[
(x + 5)^2 + (y - 2)^2 = 49
\][/tex]
