To find the correct equation representing the circle [tex]\(C\)[/tex] with center [tex]\((-2, 10)\)[/tex] and containing the point [tex]\(P(10,5)\)[/tex], we proceed as follows:
### Step 1: The Standard Form of a Circle's Equation
The standard form of a circle's equation is given by:
[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.
### Step 2: Identifying the Circle's Center
From the given information, the center of the circle [tex]\(C\)[/tex] is [tex]\((-2, 10)\)[/tex]. Therefore, in the circle's equation:
[tex]\(h = -2\)[/tex] and [tex]\(k = 10\)[/tex].
### Step 3: Determining the Radius
The radius [tex]\(r\)[/tex] can be found by calculating the distance between the center [tex]\((-2, 10)\)[/tex] and the given point [tex]\(P(10, 5)\)[/tex].
The distance formula between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Plugging in the coordinates of the center and the point [tex]\(P\)[/tex]:
[tex]\[
d = \sqrt{(10 - (-2))^2 + (5 - 10)^2}
\][/tex]
[tex]\[
d = \sqrt{(10 + 2)^2 + (5 - 10)^2}
\][/tex]
[tex]\[
d = \sqrt{12^2 + (-5)^2}
\][/tex]
[tex]\[
d = \sqrt{144 + 25}
\][/tex]
[tex]\[
d = \sqrt{169}
\][/tex]
Thus, the radius [tex]\(r\)[/tex] is [tex]\(\sqrt{169} = 13\)[/tex].
### Step 4: Writing the Equation
Now that we have the center [tex]\((-2, 10)\)[/tex] and the radius [tex]\(13\)[/tex], we can write the equation of the circle. Substituting [tex]\(h = -2\)[/tex], [tex]\(k = 10\)[/tex], and [tex]\(r = 13\)[/tex] into the standard form equation:
[tex]\[
(x - (-2))^2 + (y - 10)^2 = 13^2
\][/tex]
[tex]\[
(x + 2)^2 + (y - 10)^2 = 169
\][/tex]
### Step 5: Matching with Given Options
We look at the options provided:
A. [tex]\((x-2)^2+(y+10)^2=13\)[/tex]
B. [tex]\((x-2)^2+(y+10)^2=169\)[/tex]
C. [tex]\((x+2)^2+(y-10)^2=13\)[/tex]
D. [tex]\((x+2)^2+(y-10)^2=169\)[/tex]
The equation we derived matches exactly with option [tex]\(D\)[/tex]:
[tex]\[
(x + 2)^2 + (y - 10)^2 = 169
\][/tex]
Therefore, the correct answer is:
D. [tex]\((x+2)^2+(y-10)^2=169\)[/tex]
