To determine the equation of a circle in standard form, we need to identify the circle's center and radius. The standard form of a circle's equation is [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.
1. Determine the Center of the Circle:
- The given center of the circle is [tex]\((-5, 2)\)[/tex].
2. Calculate the Radius:
- The diameter of the circle is given as 12 units.
- The radius [tex]\(r\)[/tex] is half of the diameter: [tex]\( r = \frac{12}{2} = 6 \)[/tex].
3. Form the Equation:
- Plug the center coordinates [tex]\((h, k) = (-5, 2)\)[/tex] and the radius [tex]\(r = 6\)[/tex] into the standard form equation:
[tex]\[
(x - (-5))^2 + (y - 2)^2 = 6^2
\][/tex]
- Simplify the equation:
[tex]\[
(x + 5)^2 + (y - 2)^2 = 36
\][/tex]
4. Identify the Correct Option:
- The correct equation matching [tex]\((x + 5)^2 + (y - 2)^2 = 36\)[/tex] is presented in the fourth option.
Therefore, the correct answer is:
[tex]\[
(x + 5)^2 + (y - 2)^2 = 36
\][/tex]
