To determine the equation representing a circle with a given center and radius, we need to use the standard form of the equation of a circle:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
where [tex]\((h, k)\)[/tex] is the center of the circle and [tex]\(r\)[/tex] is the radius.
Given:
- The center of the circle is [tex]\((-3, -5)\)[/tex], so [tex]\(h = -3\)[/tex] and [tex]\(k = -5\)[/tex].
- The radius of the circle is [tex]\(6\)[/tex] units, so [tex]\(r = 6\)[/tex].
Substituting these values into the standard form equation:
[tex]\[
(x - (-3))^2 + (y - (-5))^2 = 6^2
\][/tex]
This simplifies to:
[tex]\[
(x + 3)^2 + (y + 5)^2 = 36
\][/tex]
Therefore, the equation of the circle is:
[tex]\[
(x + 3)^2 + (y + 5)^2 = 36
\][/tex]
The correct option among the given choices is:
[tex]\[
(x + 3)^2 + (y + 5)^2 = 36
\][/tex]
