Finding the Equation of a Circle with Given Center and Radius

In this article, we will explain how to find the equation of a circle given its center and radius. This information is essential for understanding the basic properties of circles and is useful in various mathematical and practical applications. We will walk through the process with a specific example.

Standard Form of the Circle Equation

The standard form of the equation of a circle is given by:

x - h^2 y - k^2 r^2

Where (h, k) are the coordinates of the center of the circle and r is the radius. This equation represents a circle centered at (h, k) with radius r.

Example: Center (3, -8) and Radius 9

Let's apply this to a specific example where the center of the circle is at (3, -8) and the radius is 9.

Step-by-Step Solution

Step 1: Identify the Center and Radius

We are given the center of the circle as (3, -8) and the radius as 9.

h 3 k -8 r 9

Step 2: Substitute into the Standard Equation

Substitute the values into the standard form of the circle equation:

x - 3^2 y - (-8)^2 9^2

This simplifies to:

x - 3^2 y 8^2 9^2

Step 3: Simplify and Write the Equation

Calculate 9^2 81.

Simplify the equation:

x - 3^2 y 8^2 81

Expanding and simplifying further:

x^2 - 6x 9 y^2 16y 64 - 81 0

Combine like terms:

x^2 - 6x y^2 16y - 8 0

Or in the standard form:

x - 3^2 y 8^2 81

So, the equation of the circle is:

x - 3^2 y - 8^2 81

This represents a circle with a center at (3, -8) and a radius of 9.

General Form of the Circle Equation

It's also useful to know the general form of the circle equation:

x^2 y^2 - 6x 16y - 8 0

Understanding the Circle Equation

When the point (3, -8) is given as the center and the radius is 9, we can use the standard form to find the equation. The equation x - 3^2 y - (-8)^2 9^2 is the correct and simplified form of the equation for the given circle.

This equation can be used to find the coordinates of any point on the circle or to determine the distance from the center to any point within the circle. It's a fundamental concept in geometry and is widely used in engineering, architecture, and various other fields.

Conclusion

By following the steps outlined in this guide, you can easily find the equation of a circle given its center and radius. Whether you need to simplify the equation or convert it to the general form, understanding the standard form and the properties of circles is essential.

Keep this guide handy for future reference and explore more related concepts to deepen your understanding of circles and their applications in mathematics and real-life scenarios.