In Exercises 107–108, write the standard form of the equation of the circle with the given center and radius. Center (-2. 4), r = 6

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 \((-2, 4)\) into the equation, replacing \(h\) with \(-2\) and \(k\) with \(4\).
3. Substitute the given radius \(r = 6\) into the equation.
4. Write the equation as \((x - (-2))^2 + (y - 4)^2 = 6^2\).
5. Simplify the equation to \((x + 2)^2 + (y - 4)^2 = 36\).

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 (-2, 4), meaning the circle is positioned 2 units left and 4 units up from the origin on the Cartesian plane. 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. In this problem, the radius is given as r = 6, which means that every point on the circle is 6 units away from the center. This value is essential for determining the right side of the standard form equation.