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The standard equation of a circle is a key formula in geometry that defines all the points on a circle. Understanding this equation is crucial when working with circles. The standard form is given by:
\[ \text{(x - h)}^2 + \text{(y - k)}^2 = r^2 \]
In this formula:

• \(h\) is the x-coordinate of the circle's center.
• \(k\) is the y-coordinate of the circle's center.
• \(r\) is the radius of the circle.

The equation essentially states that for any point \((x, y)\) on the circle, the sum of the squares of its horizontal and vertical distances from the center equals the square of the radius. By substituting the circle's center and radius into the equation, you can easily derive the specific circle's equation.

The radius of a circle is the distance from the center to any point on the circle. In the given problem, the radius is \( \frac{1}{2} \). Calculating the radius is straightforward if you know the standard equation:

• If you know the circle's equation, like \( (x + 2)^2 + (y - 5)^2 = \frac{1}{4} \), you can identify the radius by taking the square root of the constant term on the right side.
• In this example: \[ \left( \frac{1}{2} \right)^2 = \frac{1}{4}, \] so the radius is \( \frac{1}{2} \).

The radius is an important measurement because it defines the size of the circle. A larger radius means a larger circle, and vice versa. It's also essential in other calculations, such as finding the circumference or area of the circle.

The coordinates of the center of a circle provide the exact location from which all the points on the circle are equally distant. From the standard equation of a circle: \[ \text{(x - h)}^2 + \text{(y - k)}^2 = r^2 \]
The values \(h\) and \(k\) represent the center's coordinates.
In our exercise, the center is given as \((-2, 5)\). This means:

• \(h = -2\)
• \(k = 5\)

Plugging these values into the standard equation gives us a specific form that describes our circle. Specifically, the coordinates \((h, k)\) shift the circle along the x and y axes to the position \((-2, 5)\) from the origin. Understanding the center coordinates helps in graphing the circle and performing transformations, like translations, on the circle.