In Exercises 31–40, write the standard form of the equation of the circle with the given center and radius. Center (3, 2), r = 5

Verified step by step guidance

1. Identify the standard form of the equation of a circle, which is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.
2. Substitute the given center \((3, 2)\) into the equation, replacing \(h\) with 3 and \(k\) with 2.
3. Substitute the given radius \(r = 5\) into the equation.
4. Square the radius \(r\) to find \(r^2\).
5. Write the equation in standard form using the substituted values.

Key Concepts

Standard Form of a Circle's Equation

The standard form of a circle's equation is given by (x - h)² + (y - k)² = r², where (h, k) is the center of the circle and r is the radius. This format allows for easy identification of the circle's center and radius, facilitating graphing and analysis.

Coordinates of the Center

The center of a circle is represented by the coordinates (h, k). In this case, the center is (3, 2), meaning the circle is positioned 3 units along the x-axis and 2 units along the y-axis from the origin. Understanding the center's coordinates is crucial for accurately writing the equation.

Radius of a Circle

The radius of a circle is the distance from the center to any point on the circle. It is denoted by r and is essential for determining the size of the circle. In this problem, the radius is given as 5, which means the circle extends 5 units from its center in all directions.