To determine the center and diameter of the circle given by the equation [tex]\((x-7)^2+(y+3)^2=64\)[/tex], we can break down the problem step-by-step:
1. Identify the Center:
The general equation of a circle is [tex]\((x-h)^2+(y-k)^2=r^2\)[/tex], where [tex]\((h, k)\)[/tex] is the center of the circle and [tex]\(r\)[/tex] is the radius.
- From the equation [tex]\((x-7)^2+(y+3)^2=64\)[/tex], we can see that it matches the form [tex]\((x-h)^2+(y-k)^2=r^2\)[/tex].
- By comparison, [tex]\(h = 7\)[/tex] and [tex]\(k = -3\)[/tex]. Thus, the center of the circle is at [tex]\((7, -3)\)[/tex].
2. Determine the Diameter:
The diameter of a circle is twice the radius.
- From the equation [tex]\((x-7)^2+(y+3)^2=64\)[/tex], the right-hand side ([tex]\(64\)[/tex]) represents [tex]\(r^2\)[/tex].
- To find the radius ([tex]\(r\)[/tex]), we take the square root of 64, which gives us [tex]\(r = 8\)[/tex].
- The diameter of the circle is [tex]\(2 \times r = 2 \times 8 = 16\)[/tex] units.
Putting this information together:
- The center of the circle is at [tex]\( (7, -3) \)[/tex].
- The diameter of the circle is [tex]\( 16 \)[/tex] units.
So, the answers are:
The center of the circle is at [tex]\( \boxed{(7, -3)} \)[/tex].
The diameter of the circle is [tex]\( \boxed{16} \)[/tex] units.
