Equation of a Circle: Center, Radius, and Graphing

Equation of a Circle

Circles are a type of conic section, which are geometric curves defined by the intersection of a plane with a cone. The equation of a circle is a mathematical representation that describes the relationship between the center, radius, and points on the circle. In this article, we will discuss how to find the center and radius of a circle, how to write its equation in standard form, and how to graph it.

Finding the center and radius

The equation of a circle is given by the formula:

(x - h)^2 + (y - k)^2 = r^2

where (h, k) is the center of the circle and r is its radius. To find the center, you need to identify the constants immediately following x and y and flip their signs. For example, if the equation is x^2 + y^2 - 10x + 6y - 30 = 0, the center is (-5, 3).

To find the radius, you can complete the square by adding (h^2 + k^2) to both sides of the equation. Then, take the square root of the right side to find the radius. For example, if the equation is (x - 2)^2 + (y - 3)^2 = 9, the radius is 3.

Equation of a circle with a given center and radius

To find the equation of a circle with a given center and radius, you can use the formula:

(x - h)^2 + (y - k)^2 = r^2

where (h, k) is the center of the circle and r is its radius. For example, if the center is at (3, 4) and the radius is 5, the equation of the circle is (x - 3)^2 + (y - 4)^2 = 25.

Graphing circles

To graph a circle, you need to follow these steps:

1. Find the center of the circle by identifying the constants immediately following x and y and flipping their signs.
2. Find the radius by completing the square and taking the square root of the right side.
3. Plot the center point (h, k) on the coordinate plane.
4. Count the radius units horizontally and vertically from the center to find four points, and connect them with a smooth, round curve to get the graph of the circle.

For example, to graph the circle x^2 + y^2 - 10x + 6y - 30 = 0, the center is (-5, 3), and the radius is 3. Plot the center point (-5, 3) and count 3 units horizontally and 3 units vertically to find the four points (-8, 0), (2, 6), (-8, 6), and (2, 0). Connect these points with a smooth, round curve to get the graph of the circle.

In conclusion, the equation of a circle is a powerful tool for describing and understanding the properties of circles. By learning how to find the center and radius, how to write the equation in standard form, and how to graph it, you can visualize and work with circles in a variety of contexts.