To determine the equation of circle [tex]\( C \)[/tex] that has a center at [tex]\((-2, 10)\)[/tex] and contains the point [tex]\(P(10, 5)\)[/tex], follow these steps:
1. Identify the center and point on the circle:
- Center: [tex]\((-2, 10)\)[/tex]
- Point on the circle: [tex]\(P(10, 5)\)[/tex]
2. Find the radius of the circle using the distance formula between the center [tex]\((-2, 10)\)[/tex] and the point [tex]\(P(10, 5)\)[/tex]:
[tex]\[
\text{Radius} = \sqrt{(10 - (-2))^2 + (5 - 10)^2}
\][/tex]
[tex]\[
= \sqrt{(10 + 2)^2 + (5 - 10)^2}
\][/tex]
[tex]\[
= \sqrt{12^2 + (-5)^2}
\][/tex]
[tex]\[
= \sqrt{144 + 25}
\][/tex]
[tex]\[
= \sqrt{169}
\][/tex]
[tex]\[
= 13
\][/tex]
3. Use the standard form of the equation of a 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]
Here, the center [tex]\((h, k)\)[/tex] is [tex]\((-2, 10)\)[/tex] and the radius [tex]\(r\)[/tex] is [tex]\(13\)[/tex].
4. Plug the center and radius into the standard form:
- Center: [tex]\((-2, 10)\)[/tex]
- Radius: [tex]\(13\)[/tex]
[tex]\[
(x - (-2))^2 + (y - 10)^2 = 13^2
\][/tex]
[tex]\[
(x + 2)^2 + (y - 10)^2 = 169
\][/tex]
So, the correct equation representing circle [tex]\( C \)[/tex] is:
[tex]\[
\boxed{(x+2)^2+(y-10)^2=169}
\][/tex]
Thus, the correct answer is D.
