Sure! Let's solve this step by step.
The general equation for a circle in the coordinate plane with the center [tex]\((h, k)\)[/tex] and radius [tex]\(r\)[/tex] is given by:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
Here, the center of the circle is given as [tex]\((-6, -8)\)[/tex]. Substituting [tex]\(h = -6\)[/tex] and [tex]\(k = -8\)[/tex] into the general equation, we get:
[tex]\[
(x + 6)^2 + (y + 8)^2 = r^2
\][/tex]
Next, we need to determine the radius [tex]\(r\)[/tex]. To do this, we use the point [tex]\((0, -8)\)[/tex], which lies on the circle.
Substitute [tex]\(x = 0\)[/tex] and [tex]\(y = -8\)[/tex] into the equation:
[tex]\[
(0 + 6)^2 + (-8 + 8)^2 = r^2
\][/tex]
This simplifies to:
[tex]\[
6^2 + 0^2 = r^2
\][/tex]
[tex]\[
36 + 0 = r^2
\][/tex]
[tex]\[
r^2 = 36
\][/tex]
So, the equation of the circle is:
[tex]\[
(x + 6)^2 + (y + 8)^2 = 36
\][/tex]
Among the given options, we must find the equation that matches this form. The options are:
1. [tex]\((x-6)^2 + (y-8)^2 = 6\)[/tex]
2. [tex]\((x-6)^2 + (y-8)^2 = 36\)[/tex]
3. [tex]\((x+6)^2 + (y+8)^2 = 6\)[/tex]
4. [tex]\((x+6)^2 + (y+8)^2 = 36\)[/tex]
The correct option is:
[tex]\[
(x + 6)^2 + (y + 8)^2 = 36
\][/tex]
Therefore, the correct answer is the fourth option:
[tex]\[
(x+6)^2 + (y+8)^2 = 36
\][/tex]
