To find the equation of a circle with a given center and a point through which it passes, we can follow these steps:
1. Identify the center of the circle: The center is given as [tex]\((-2, -4)\)[/tex].
2. Identify a point on the circle: The point is given as [tex]\((3, 8)\)[/tex].
3. Calculate the radius of the circle: The radius is the distance between the center [tex]\((-2, -4)\)[/tex] and the point [tex]\((3, 8)\)[/tex]. We use the distance formula to compute this distance.
[tex]\[
\text{Distance (radius)} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Substituting the given coordinates:
[tex]\[
\text{radius} = \sqrt{(3 - (-2))^2 + (8 - (-4))^2}
\][/tex]
Simplifying inside the square root:
[tex]\[
\text{radius} = \sqrt{(3 + 2)^2 + (8 + 4)^2} = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13
\][/tex]
4. Write the equation of the circle: The standard form for the equation of a circle is:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
Here, [tex]\((h, k)\)[/tex] is the center [tex]\((-2, -4)\)[/tex], and [tex]\(r\)[/tex] is the radius [tex]\(13\)[/tex].
5. Substitute the center coordinates and radius into the circle equation:
[tex]\[
(x - (-2))^2 + (y - (-4))^2 = 13^2
\][/tex]
Simplifying:
[tex]\[
(x + 2)^2 + (y + 4)^2 = 169
\][/tex]
Therefore, the equation of the circle is:
[tex]\[
(x + 2)^2 + (y + 4)^2 = 169
\][/tex]
The correct answer from the provided options is:
[tex]\[
(x+2)^2+(y+4)^2=169
\][/tex]
