How to Determine the Equation of a Circle with a Radius of 6 and Center (3, 5): A Comprehensive Guide

The equation of a circle is a fundamental concept in mathematics, essential for understanding geometry and its applications in various fields such as engineering, physics, and computer graphics. This guide will walk you through the process of determining the equation of a circle with a specified radius and center point. Specifically, we will find the equation of a circle with a radius of 6 and a center at (3, 5).

Understanding the Circle Equation

The standard form of the equation of a circle with the center at ((h, k)) and radius (r) is:

[ (x - h)^2 (y - k)^2 r^2 ]

This equation represents all points in the plane that are at a distance (r) from the center ((h, k)). Let's break this down further:

((x - h)^2) is the square of the distance in the x-direction from the center to the point on the circle. ((y - k)^2) is the square of the distance in the y-direction from the center to the point on the circle. (r^2) is the square of the radius, representing the total distance from the center to any point on the circumference.

Deriving the Equation for a Specific Circle

To find the equation of a circle with a center at ((3, 5)) and a radius of 6, we follow a straightforward process:

1. Substitute (h, k,) and (r)

Here, we have (h 3), (k 5), and (r 6).

2. Write the equation:

[ (x - 3)^2 (y - 5)^2 6^2 ]

3. Simplify the right-hand side

[ (x - 3)^2 (y - 5)^2 36 ]

This is the equation of the circle with a center at ((3, 5)) and a radius of 6. This equation tells us that any point ((x, y)) on the circle is 6 units away from the center ((3, 5)).

Using the General Form of the Circle Equation

Alternatively, we can use the general form of the circle equation:

[ x - a^2 y - b^2 r^2 ]

To find the equation, set (a 3), (b 5), and (r 6), giving us:

[ x - 3^2 y - 5^2 6^2 ]

Simplifying, we get:

[ x - 9 y - 25 36 ]

[ x - 3^2 y - 5^2 36 ]

What About a Circle with Center (3, 4) and Radius 6?

To find the equation of a circle with a center at ((3, 4)) and a radius of 6, we follow the same process:

1. Substitute (h, k,) and (r)

Here, we have (h 3), (k 4), and (r 6).

2. Write the equation:

[ (x - 3)^2 (y - 4)^2 6^2 ]

3. Simplify the right-hand side

[ (x - 3)^2 (y - 4)^2 36 ]

This is the equation of the circle with a center at ((3, 4)) and a radius of 6.

Conclusion

Understanding how to determine the equation of a circle is crucial for solving a variety of mathematical problems. By following the steps provided in this guide, you can easily derive the equation of a circle given its center and radius. This knowledge is valuable in many fields, from pure mathematics to practical applications in engineering and science.