To determine the center of a circle given its equation in standard form, we need to understand the general form of the equation of a circle, which is [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex]. In this equation, [tex]\((h, k)\)[/tex] represents the center of the circle, and [tex]\(r\)[/tex] is the radius.
Given the equation of the circle:
[tex]\[
(x + 4)^2 + (y - 7)^2 = 16
\][/tex]
We can rewrite the equation in a form that matches the general form. Notice that:
- [tex]\( (x + 4)^2 \)[/tex] can be written as [tex]\( (x - (-4))^2 \)[/tex]
- [tex]\( (y - 7)^2 \)[/tex] is already in the correct form [tex]\( (y - 7)^2 \)[/tex]
By comparing the given equation [tex]\((x + 4)^2 + (y - 7)^2 = 16\)[/tex] with the general form [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex], we can identify [tex]\(h\)[/tex] and [tex]\(k\)[/tex] as follows:
- Since [tex]\(x + 4\)[/tex] matches [tex]\((x - (-4))\)[/tex], we can see that [tex]\(h = -4\)[/tex]
- Since [tex]\(y - 7\)[/tex] matches [tex]\((y - 7)\)[/tex], we see that [tex]\(k = 7\)[/tex]
Therefore, the center of the circle is [tex]\((-4, 7)\)[/tex].
Hence, the correct answer is:
[tex]\[
(-4, 7)
\][/tex]
