To determine the equation of the circle [tex]\( C \)[/tex] with a center at [tex]\( (-2, 10) \)[/tex] and containing the point [tex]\( P(10, 5) \)[/tex], we need to follow these steps:
1. Identify the center of the circle:
The center of the circle is given as [tex]\( (h, k) = (-2, 10) \)[/tex].
2. Identify a point on the circle:
The point on the circle is [tex]\( P(10, 5) \)[/tex].
3. Calculate the radius of the circle:
The radius [tex]\( r \)[/tex] can be calculated using the distance formula between the center [tex]\((h, k)\)[/tex] and a point [tex]\((x_1, y_1)\)[/tex] on the circle:
[tex]\[
r = \sqrt{(x_1 - h)^2 + (y_1 - k)^2}
\][/tex]
Substituting the given points:
[tex]\[
r = \sqrt{(10 - (-2))^2 + (5 - 10)^2}
\][/tex]
Simplifying inside the square root:
[tex]\[
r = \sqrt{(10 + 2)^2 + (5 - 10)^2}
\][/tex]
[tex]\[
r = \sqrt{12^2 + (-5)^2}
\][/tex]
[tex]\[
r = \sqrt{144 + 25}
\][/tex]
[tex]\[
r = \sqrt{169}
\][/tex]
[tex]\[
r = 13
\][/tex]
4. Write 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]
Substituting [tex]\( h = -2 \)[/tex], [tex]\( k = 10 \)[/tex], and [tex]\( r = 13 \)[/tex]:
[tex]\[
(x - (-2))^2 + (y - 10)^2 = 13^2
\][/tex]
Simplifying the equation:
[tex]\[
(x + 2)^2 + (y - 10)^2 = 169
\][/tex]
Therefore, the correct equation that represents the circle [tex]\( C \)[/tex] is:
[tex]\[
(x + 2)^2 + (y - 10)^2 = 169
\][/tex]
