To determine the equation of a circle, we use the standard form of the equation, which is:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
where [tex]\((h, k)\)[/tex] represents the center of the circle and [tex]\(r\)[/tex] is the radius.
Given the problem:
- The center of the circle is [tex]\((-3, -5)\)[/tex].
- The radius of the circle is [tex]\(4\)[/tex].
Let's substitute these values into the standard equation.
1. Identify the center [tex]\((h, k)\)[/tex]:
Here, [tex]\(h = -3\)[/tex] and [tex]\(k = -5\)[/tex].
2. Identify the radius [tex]\(r\)[/tex]:
Here, [tex]\(r = 4\)[/tex].
3. Substitute values into the standard form:
[tex]\[
(x - (-3))^2 + (y - (-5))^2 = 4^2
\][/tex]
4. Simplify the minus signs:
[tex]\[
(x + 3)^2 + (y + 5)^2 = 16
\][/tex]
Therefore, the equation of the circle is:
[tex]\[
(x + 3)^2 + (y + 5)^2 = 16
\][/tex]
Now, let's match this equation with the given options.
A. [tex]\((x - 3)^2 + (y - 5)^2 = 16\)[/tex]
B. [tex]\((x + 3)^2 + (y + 5)^2 = 16\)[/tex]
C. [tex]\((x + 3)^2 + (y + 5)^2 = 4\)[/tex]
D. [tex]\((x - 3)^2 + (y - 5)^2 = 4\)[/tex]
The correct option is B:
[tex]\[
(x + 3)^2 + (y + 5)^2 = 16
\][/tex]
So, the correct answer is option B.
