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Finding the center of a circle in an equation like \( \left(x+\frac{1}{2}\right)^{2}+\left(y+\frac{1}{3}\right)^{2}=\frac{16}{25} \) involves comparing it to the standard circle equation \((x-h)^2 + (y-k)^2 = r^2\). The center of a circle is denoted by the coordinates \((h, k)\).
To determine these values:

• Identify the expressions within the parentheses: \(x+\frac{1}{2}\) and \(y+\frac{1}{3}\).
• Set them equal to \(x-h\) and \(y-k\), respectively.

Since the equation uses additions, we find:

• \(h\) is the opposite number of \(+\frac{1}{2}\), thus \(h = -\frac{1}{2}\).
• \(k\) is the opposite of \(+\frac{1}{3}\), resulting in \(k = -\frac{1}{3}\).

Therefore, the center of the circle is at \((-\frac{1}{2}, -\frac{1}{3})\). This method is straightforward because whatever values appear with \(x\) and \(y\) come from completing the square. Simply put, the center is the negative value of these constants, which indicates the shift from the origin in both the x and y directions.

The radius of a circle is derived from the equation \((x-h)^2 + (y-k)^2 = r^2\), specifically from the term \(r^2\) on the right-hand side. In the given equation, \(r^2 = \frac{16}{25}\), this means \(r\) is the square root of \(\frac{16}{25}\). Calculating this involves straightforward steps:

• Take the square root of \(\frac{16}{25}\).
• This breaks down into \(\sqrt{16}/\sqrt{25}\), which equals \(\frac{4}{5}\).

The process shows the radius of the circle is \(\frac{4}{5}\).
Visualizing this, the radius represents the constant distance from the circle's center \((-\frac{1}{2}, -\frac{1}{3})\) to any point on the circle's edge. It's a fundamental characteristic that helps define the circle's size. Understanding this ensures a comprehensive grasp of what the radius signifies in the circle's equation and how it impacts circle size in a geometric context.

The equation of a circle is a precise representation of all the points equidistant from a central point \((h, k)\). This standard form is \((x-h)^2 + (y-k)^2 = r^2\). By understanding this form, you can quickly identify the circle's center and radius without additional calculations.
Key takeaways:

• \((h, k)\) indicates the circle's center, showing its exact location on the coordinate plane.
• \(r\) represents the radius, which is always a positive number, indicating the distance from the center to the circle's edge.

When faced with any circle equation:

• Recognize the structure where '\(-\)' signals a positive center value and '\(+\)' signifies a negative one.
• Use the square terms \((x-h)^2\) and \((y-k)^2\) to uncover the center while \(r^2\) reveals the radius.

This form of the equation is valuable for graphing purposes or solving problems related to circle dimensions and placement. Understanding this form allows one to visualize and manipulate circle equations successfully in mathematics or real-world applications.