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. Recall the standard form of the equation of a circle: $(x\; -\; h)^2\; +\; (y\; -\; k)^2\; =\; r^2$, where $(h,\; k)$ is the center of the circle and $r$ is the radius.
2. Substitute the given center $(-2,\; 4)$ into the formula. This means $h\; =\; -2$ and $k\; =\; 4$.
3. Substitute the given radius $r\; =\; 6$ into the formula. Remember to square the radius, so $r^2\; =\; 6^2$.
4. Write the equation by replacing $h$, $k$, and $r^2$ in the standard form: $(x\; -\; (-2))^2\; +\; (y\; -\; 4)^2\; =\; 6^2$.
5. Simplify the equation: $(x\; +\; 2)^2\; +\; (y\; -\; 4)^2\; =\; 36$. This is the standard form of the equation of the circle.

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 given as (-2, 4), meaning the circle is centered at the point where x = -2 and y = 4 on the Cartesian plane. Understanding the center's coordinates is crucial for correctly applying them in the standard form 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 a critical component in the standard form equation. In this problem, the radius is given as 6, which means that the circle extends 6 units from the center in all directions.