Let's determine the equation of a circle based on the given parameters:
1. Identify the center and radius of the circle:
- Center of the circle: [tex]\((-3, -3)\)[/tex]
- Radius of the circle: 7
2. Recall 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.
3. Substitute the given center and radius into the formula:
- [tex]\(h = -3\)[/tex]
- [tex]\(k = -3\)[/tex]
- [tex]\(r = 7\)[/tex]
4. Substitute these values into the equation:
[tex]\[
(x - (-3))^2 + (y - (-3))^2 = 7^2
\][/tex]
5. Simplify the equation by addressing the signs and calculating the square of the radius:
[tex]\[
(x + 3)^2 + (y + 3)^2 = 49
\][/tex]
Therefore, the equation of the circle is:
[tex]\[
(x + 3)^2 + (y + 3)^2 = 49
\][/tex]
6. Compare this equation with the given options to select the correct one:
- Option a: [tex]\((x-3)^2+(y-3)^2=81\)[/tex]
- Option b: [tex]\((x-3)^2+(y-7)^2=25\)[/tex]
- Option c: [tex]\((x+3)^2+(y+3)^2=49\)[/tex]
- Option d: [tex]\((x-3)^2+(y+3)^2=64\)[/tex]
The correct equation matches option c:
[tex]\[
(x+3)^2+(y+3)^2=49
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{c}
\][/tex]
