Certainly! Let's analyze the given equation of the circle step-by-step:
The given equation of the circle is:
[tex]\[
(x + 4)^2 + (y - 7)^2 = 16
\][/tex]
This equation is in the standard form of a circle's equation:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
where [tex]\((h, k)\)[/tex] represents the center of the circle and [tex]\(r\)[/tex] is the radius.
In our given equation, we can identify the following:
- [tex]\((x + 4)\)[/tex] can be rewritten as [tex]\((x - (-4))\)[/tex].
- [tex]\((y - 7)\)[/tex] remains the same.
From this, we can see that the center of the circle [tex]\((h, k)\)[/tex] is:
[tex]\[
(-4, 7)
\][/tex]
Also, we can identify that the [tex]\(r^2\)[/tex] term is:
[tex]\[
16
\][/tex]
To find the radius [tex]\(r\)[/tex], we need to take the square root of [tex]\(16\)[/tex]:
[tex]\[
r = \sqrt{16}
\][/tex]
Calculating the square root, we get:
[tex]\[
r = 4
\][/tex]
Therefore, the radius of the circle is:
[tex]\[
\boxed{4}
\][/tex]
