To determine the center of the circle given by the equation [tex]\((x-14)^2 + (y+21)^2 = 64\)[/tex], we need to recall the standard form of the equation of a circle in a plane.
The standard form of the circle equation is
[tex]\[
(x-h)^2 + (y-k)^2 = r^2
\][/tex]
where [tex]\((h, k)\)[/tex] represents the coordinates of the center of the circle and [tex]\(r\)[/tex] represents the radius of the circle.
Given the equation [tex]\((x-14)^2 + (y+21)^2 = 64\)[/tex], we can compare it directly with the standard form [tex]\((x-h)^2 + (y-k)^2 = r^2\)[/tex].
By comparing:
- The term [tex]\((x-14)^2\)[/tex] suggests that [tex]\(h = 14\)[/tex].
- The term [tex]\((y+21)^2\)[/tex] suggests that [tex]\(k = -21\)[/tex].
Therefore, the coordinates of the center of the circle are [tex]\((14, -21)\)[/tex].
Among the options given:
1. [tex]\((-14, 21)\)[/tex]
2. [tex]\((14, 21)\)[/tex]
3. [tex]\((14, -21)\)[/tex]
4. [tex]\((-14, -21)\)[/tex]
The correct answer is [tex]\((14, -21)\)[/tex], which corresponds to option 3.
