Verified Answer :

  • The equation of a circle is \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the center.
  • By matching the given equation \((x-5)^2 + (y+6)^2 = 4^2\) with the standard form, we determine that \(h=5\) and \(k=-6\).
  • The center of the circle is \((5, -6)\).
  • Final Answer: \(\boxed{(5, -6)}\).

Explanation

  1. Analyzing the standard form of a circle equation.

The equation of a circle in standard form is given by \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) represents the coordinates of the center of the circle and \(r\) is the radius. The problem provides the specific equation \((x-5)^2 + (y+6)^2 = 4^2\). By comparing this with the standard form, we can identify the values for \(h\) and \(k\).

  1. Extracting the center coordinates by comparing equations.

To find the center \((h, k)\), we compare each part of the given equation to the standard form:

  1. The term \((x-5)^2\) corresponds to \((x-h)^2\), which gives \(h = 5\).
  2. The term \((y+6)^2\) can be written as \((y - (-6))^2\), which corresponds to \((y-k)^2\), giving \(k = -6\).

Thus, the center of the circle is \((h, k) = (5, -6)\).

  1. Identifying the correct option and concluding the result.

Based on our analysis, the center of the circle defined by the equation \((x-5)^2 + (y+6)^2 = 4^2\) is the point \((5, -6)\). Comparing this to the given options:

  • \((-6, 5)\)
  • \((-5, 6)\)
  • \((5, -6)\)
  • \((6, -5)\)

The correct choice is \((5, -6)\). Therefore, the final answer is \(\boxed{(5, -6)}\).

Examples

Knowing how to identify the center of a circle from its equation is fundamental in fields like computer graphics and structural engineering. For example, when designing a circular support structure in a CAD program, an engineer needs to define the center point to place the object correctly within the workspace. By translating the equation of the circle into its center and radius, the engineer ensures the structure is perfectly positioned relative to other components, allowing for precise and efficient design.