Which is the equation for a circle with center at [tex]$(-2, -4)$[/tex] that passes through the point [tex]$(3, 8)$[/tex]?

[tex]
\begin{array}{l}
\text{A. } (x+2)^2 + (y+4)^2 = 169 \\
\text{B. } (x-2)^2 + (y+4)^2 = 169 \\
\text{C. } (x+2)^2 + (y+4)^2 = 144 \\
\text{D. } (x-2)^2 + (y+4)^2 = 144
\end{array}
[/tex]



Answer :

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]