To determine the center and diameter of the circle given by the equation [tex]\((x-7)^2+(y+3)^2=64\)[/tex], we need to recall the standard form of a circle's equation:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
Here, [tex]\((h, k)\)[/tex] represents the center of the circle, and [tex]\(r\)[/tex] is the radius of the circle.
1. Identifying the center:
- By comparing the given equation [tex]\((x-7)^2+(y+3)^2=64\)[/tex] with the standard form [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex], we can see that:
- [tex]\(h = 7\)[/tex]
- [tex]\(k = -3\)[/tex]
- Therefore, the center of the circle is [tex]\((7, -3)\)[/tex].
2. Determining the diameter:
- The right-hand side of the equation represents [tex]\(r^2\)[/tex], which in this case is [tex]\(64\)[/tex].
- To find the radius [tex]\(r\)[/tex], we take the square root of [tex]\(64\)[/tex]:
[tex]\[
r = \sqrt{64} = 8
\][/tex]
- The diameter [tex]\(D\)[/tex] of the circle is twice the radius:
[tex]\[
D = 2 \times r = 2 \times 8 = 16 \text{ units}
\][/tex]
After following these steps, the center of the circle is at [tex]\((7, -3)\)[/tex] and the diameter is [tex]\(16\)[/tex] units.
So, the correct answers are:
- The center of the circle is at [tex]\((7, -3)\)[/tex].
- The diameter of the circle is [tex]\(16\)[/tex] units.
