Select the correct answer.

What is the equation in standard form for a circle with center at [tex]$(-5, 2)$[/tex] and a diameter of 12 units?

A. [tex]$(x-5)^2+(y+2)^2=36$[/tex]

B. [tex]$(x+5)^2+(y-2)^2=144$[/tex]

C. [tex]$(x-5)^2+(y+2)^2=144$[/tex]

D. [tex]$(x+5)^2+(y-2)^2=36$[/tex]



Answer :

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]