To identify the equation of the circle that has its center at [tex]\((9, 12)\)[/tex] and passes through the origin [tex]\((0, 0)\)[/tex], follow these steps:
1. Determine the Center and a Point on the Circle:
Given:
- Center of the circle [tex]\((h, k)\)[/tex] is [tex]\((9, 12)\)[/tex].
- The circle passes through the origin [tex]\((0, 0)\)[/tex].
2. Calculate the Radius of the Circle:
The radius [tex]\(r\)[/tex] of the circle can be found using the distance formula between the center [tex]\((9, 12)\)[/tex] and the origin [tex]\((0, 0)\)[/tex]:
[tex]\[
r = \sqrt{(x_1 - h)^2 + (y_1 - k)^2}
\][/tex]
Substitute the values:
[tex]\[
r = \sqrt{(0 - 9)^2 + (0 - 12)^2} = \sqrt{81 + 144} = \sqrt{225} = 15
\][/tex]
3. Formulate the Equation of the Circle:
The standard form of the equation of a circle with center [tex]\((h, k)\)[/tex] and radius [tex]\(r\)[/tex] is:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
Substitute [tex]\(h = 9\)[/tex], [tex]\(k = 12\)[/tex], and [tex]\(r = 15\)[/tex]:
[tex]\[
(x - 9)^2 + (y - 12)^2 = 15^2
\][/tex]
Which simplifies to:
[tex]\[
(x - 9)^2 + (y - 12)^2 = 225
\][/tex]
4. Identify the Correct Equation from Given Options:
Among the given options, the correct equation is:
[tex]\[
\text{D.}\ (x - 9)^2 + (y - 12)^2 = 225
\][/tex]
Hence, the equation of the circle that has its center at [tex]\((9, 12)\)[/tex] and passes through the origin is:
Option D. [tex]\((x - 9)^2 + (y - 12)^2 = 225\)[/tex]
