To find the equation of a circle with a given center and a point that lies on the circle, we need to determine the radius of the circle first and then use the standard form of the equation of a circle.
### Step-by-Step Solution:
1. Identify the center and the given point:
- The center of the circle is [tex]\((-2, -4)\)[/tex].
- The point on the circle is [tex]\((3, 8)\)[/tex].
2. Calculate the radius:
- The radius [tex]\( r \)[/tex] is the distance from the center to the given point.
- We use the distance formula to find the radius:
[tex]\[
r = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}
\][/tex]
Substituting the given values:
[tex]\[
r = \sqrt{(3 - (-2))^2 + (8 - (-4))^2}
\][/tex]
[tex]\[
r = \sqrt{(3 + 2)^2 + (8 + 4)^2}
\][/tex]
[tex]\[
r = \sqrt{5^2 + 12^2}
\][/tex]
[tex]\[
r = \sqrt{25 + 144}
\][/tex]
[tex]\[
r = \sqrt{169}
\][/tex]
[tex]\[
r = 13
\][/tex]
3. 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]
- For our given center [tex]\((-2, -4)\)[/tex] and radius [tex]\( 13 \)[/tex]:
[tex]\[
(x - (-2))^2 + (y - (-4))^2 = 13^2
\][/tex]
[tex]\[
(x + 2)^2 + (y + 4)^2 = 169
\][/tex]
### Conclusion:
The equation of the circle with center [tex]\((-2, -4)\)[/tex] that passes through the point [tex]\((3, 8)\)[/tex] is:
[tex]\[
(x + 2)^2 + (y + 4)^2 = 169
\][/tex]
Therefore, the correct option is:
[tex]\[
(x + 2)^2 + (y + 4)^2 = 169
\][/tex]
