How to Find the Equation of a Circle Given its Center

When working with circles in mathematics, particularly in geometry and algebra, it is often necessary to find the equation of a circle given its center and radius. This article will guide you through the process and provide several examples to illustrate the steps involved.

Standard Equation of a Circle

The standard equation of a circle is given by:

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

Where:

h and k are the coordinates of the center of the circle.

r is the radius of the circle.

Steps to Write the Equation of a Circle

To write the equation of a circle when you know its center and radius, follow these steps:

Identify the Center:

Determine the coordinates of the center of the circle, denoted as (h, k).

Determine the Radius:

Obtain the radius of the circle, denoted as r.

Substitute into the Equation:

Plug the values of h, k, and r into the standard equation of a circle.

Example: Finding the Equation of a Circle

Suppose we have a circle with center (3, -2) and radius 5. Let's walk through the process of finding its equation.

Identify the Center:

The center is (3, -2). So, h 3 and k -2.

Determine the Radius:

The radius is given as 5.

Substitute into the Equation:

Substitute h 3, k -2, and r 5 into the standard equation:

x - 3^2 y - (-2)^2 5^2

Simplify:

The equation simplifies to:

(x - 3)^2 (y 2)^2 25

This is the equation of the circle with center (3, -2) and radius 5.

Alternative Derivation: Using the Distance Formula

Another way to derive the equation of a circle is by using the distance formula. According to the distance formula:

sqrt{(x - a)^2 (y - b)^2} r

Squaring both sides gives:

(x - a)^2 (y - b)^2 r^2

Where (a, b) are the center coordinates, and r is the radius of the circle. This is the same as the standard equation of a circle.

Conclusion

To sum up, finding the equation of a circle given its center and radius can be achieved through the standard equation or by using the distance formula. Understanding these methods will help you work with circles more effectively in various mathematical applications.

For further reading on this topic, check out the following video: