To find which equation represents a circle with a center at [tex]\( (-3, -5) \)[/tex] and a radius of 6 units, we need to use the standard form of the equation for a circle. The general form of the equation for a circle with center [tex]\( (h, k) \)[/tex] and radius [tex]\( r \)[/tex] is given by:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
Given:
- The center [tex]\((h, k) = (-3, -5)\)[/tex]
- The radius [tex]\(r = 6\)[/tex]
We can substitute these values into the standard form of the equation:
1. Substitute [tex]\( h = -3 \)[/tex] into the equation:
[tex]\[
(x - (-3))^2 + (y - k)^2 = r^2
\][/tex]
This simplifies to:
[tex]\[
(x + 3)^2 + (y - k)^2 = r^2
\][/tex]
2. Substitute [tex]\( k = -5 \)[/tex] into the equation:
[tex]\[
(x + 3)^2 + (y - (-5))^2 = r^2
\][/tex]
This simplifies to:
[tex]\[
(x + 3)^2 + (y + 5)^2 = r^2
\][/tex]
3. Substitute [tex]\( r = 6 \)[/tex] into the equation:
[tex]\[
(x + 3)^2 + (y + 5)^2 = 6^2
\][/tex]
Since [tex]\( 6^2 = 36 \)[/tex], the equation further simplifies to:
[tex]\[
(x + 3)^2 + (y + 5)^2 = 36
\][/tex]
Therefore, the correct equation representing the circle with a center at [tex]\( (-3, -5) \)[/tex] and a radius of 6 units is:
[tex]\[
(x + 3)^2 + (y + 5)^2 = 36
\][/tex]
Hence, the equation is the fourth option on the list:
[tex]\[
\boxed{(x + 3)^2 + (y + 5)^2 = 36}
\][/tex]
