Finding The Equation Of A Circle With Diameter Endpoints (-3,-10) And (4,-6)

Jul 11, 2025 by Admin 77 views

Finding the Equation of a Circle Given the Endpoints of a Diameter

In the realm of mathematics, particularly geometry, circles hold a fundamental position. Understanding their properties and equations is crucial for various applications. One common problem involves determining the equation of a circle when given the endpoints of its diameter. This article delves into a step-by-step approach to solve this problem, providing a comprehensive guide for students and enthusiasts alike. This article aims to explore how to find the equation of a circle when provided with the endpoints of a diameter, offering a detailed, step-by-step solution suitable for students and math enthusiasts. By the end of this guide, readers will be equipped with the knowledge and skills to tackle similar problems with confidence. Understanding the properties and equations of circles is fundamental in geometry and has various practical applications, making this a valuable skill to acquire. Let's delve into the process of finding the equation by understanding the underlying concepts and applying them methodically.

Understanding the Circle Equation

Before we dive into the problem-solving process, let's first establish a solid understanding of the equation of a circle. The standard equation of a circle with center $(h, k)$ and radius $r$ is given by:

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

Where:

$(x, y)$ represents any point on the circumference of the circle.
$(h, k)$ represents the coordinates of the center of the circle.
$r$ represents the radius of the circle.

This equation is derived from the Pythagorean theorem, where the distance between any point on the circle and the center is equal to the radius. To find the equation of a circle, we need to determine the values of $h$ , $k$ , and $r$ . Given the endpoints of a diameter, we can find the center by calculating the midpoint of the diameter, and the radius by finding half the length of the diameter. This foundational knowledge is crucial for approaching our specific problem. This section emphasizes the importance of grasping the standard equation of a circle, which is pivotal for solving problems related to circles. The equation $(x - h)^2 + (y - k)^2 = r^2$ is the cornerstone of our approach. We'll break down each component of the equation – $(x, y)$ as a point on the circle, $(h, k)$ as the center, and $r$ as the radius. Understanding how this equation is derived from the Pythagorean theorem will provide a deeper insight into its applications. We'll discuss how the center and radius are essential parameters that define a circle, and how knowing these values allows us to fully describe the circle mathematically. The ability to manipulate and apply this equation is a fundamental skill in coordinate geometry, and this section aims to solidify that understanding. Let's further explore how this equation can be applied in the context of our given problem.

Problem Statement: Diameter Endpoints and the Circle Equation

Our specific problem states that the diameter of a circle has endpoints at $(-3, -10)$ and $(4, -6)$ . The goal is to find the equation of this circle. To achieve this, we will need to follow these key steps:

Find the Center: Determine the midpoint of the diameter, which will give us the center $(h, k)$ of the circle.
Find the Radius: Calculate the distance between the center and one of the endpoints of the diameter. This distance will be the radius $r$ of the circle.
Write the Equation: Substitute the values of $h$ , $k$ , and $r$ into the standard equation of a circle, $(x - h)^2 + (y - k)^2 = r^2$ .

This structured approach will allow us to systematically solve the problem and arrive at the equation of the circle. This section clearly lays out the problem we intend to solve: finding the equation of a circle given the endpoints of its diameter. The specified endpoints are $(-3, -10)$ and $(4, -6)$ . Understanding the problem statement is the first critical step in any mathematical endeavor. We'll emphasize the importance of visualizing the problem geometrically, picturing a circle with the given diameter endpoints. To tackle this problem effectively, we'll break it down into three manageable steps: finding the center of the circle, determining the radius, and finally, constructing the equation of the circle. This systematic approach ensures clarity and reduces the likelihood of errors. By outlining the steps explicitly, we provide a roadmap for solving the problem, making it easier for readers to follow along and understand the logical flow. Now, let's dive into the first step: finding the center of the circle using the midpoint formula.

Step 1: Finding the Center of the Circle

The center of the circle is the midpoint of its diameter. The midpoint formula is given by:

$Midpoint = igg( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \bigg)$

Where $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of the endpoints of the diameter. In our case, the endpoints are $(-3, -10)$ and $(4, -6)$ . Substituting these values into the midpoint formula, we get:

$h = \frac{-3 + 4}{2} = \frac{1}{2}$ $k = \frac{-10 + (-6)}{2} = \frac{-16}{2} = -8$

Therefore, the center of the circle is $\bigg( \frac{1}{2}, -8 \bigg)$ .

The midpoint formula plays a pivotal role in determining the center of the circle, and this section provides a thorough explanation of its application. The formula, $Midpoint = \bigg( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \bigg)$ , is presented clearly, with each term defined for better understanding. We then demonstrate how to apply this formula to the given endpoints $(-3, -10)$ and $(4, -6)$ . The step-by-step calculation of the x-coordinate, $h$ , and the y-coordinate, $k$ , of the center is shown, making it easy for readers to follow the arithmetic. This detailed calculation not only provides the answer but also reinforces the method itself. The final result, the center of the circle being $\bigg( \frac{1}{2}, -8 \bigg)$ , is clearly stated, setting the stage for the next step in the problem-solving process. This section emphasizes the importance of accurate calculation and careful substitution to arrive at the correct center coordinates. Now, let's move on to the next crucial step: determining the radius of the circle.

Step 2: Finding the Radius of the Circle

The radius of the circle is the distance between the center and any point on the circle's circumference. Since we know the center $\bigg( \frac{1}{2}, -8 \bigg)$ and the endpoints of the diameter, we can use the distance formula to find the radius. The distance formula between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

$Distance = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Let's use one of the endpoints, say $(4, -6)$ , and the center $\bigg( \frac{1}{2}, -8 \bigg)$ , to calculate the radius:

$r = \sqrt{\bigg(4 - \frac{1}{2}\bigg)^2 + (-6 - (-8))^2}$ $r = \sqrt{\bigg(\frac{7}{2}\bigg)^2 + (2)^2}$ $r = \sqrt{\frac{49}{4} + 4}$ $r = \sqrt{\frac{49 + 16}{4}}$ $r = \sqrt{\frac{65}{4}}$ $r = \frac{\sqrt{65}}{2}$

Thus, the radius of the circle is $\frac{\sqrt{65}}{2}$ .

This section provides a comprehensive guide to calculating the radius of the circle, a crucial step towards finding the circle's equation. The importance of the radius as the distance between the center and any point on the circle's circumference is re-emphasized. The distance formula, $Distance = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ , is presented clearly, setting the foundation for the calculation. We then walk through the application of the distance formula using the center $\bigg( \frac{1}{2}, -8 \bigg)$ and one of the endpoints $(4, -6)$ . Each step of the calculation is shown in detail, from substituting the values to simplifying the expression, making it easy for readers to follow along. The arithmetic is broken down into manageable parts, reducing the complexity and minimizing the chance of errors. The final result, the radius of the circle being $\frac{\sqrt{65}}{2}$ , is clearly stated. Now that we have both the center and the radius, we are ready to write the equation of the circle.

Step 3: Writing the Equation of the Circle

Now that we have determined the center $\bigg( \frac{1}{2}, -8 \bigg)$ and the radius $\frac{\sqrt{65}}{2}$ , we can substitute these values into the standard equation of a circle:

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

Substituting $h = \frac{1}{2}$ , $k = -8$ , and $r = \frac{\sqrt{65}}{2}$ , we get:

$\bigg(x - \frac{1}{2}\bigg)^2 + (y - (-8))^2 = \bigg(\frac{\sqrt{65}}{2}\bigg)^2$ $\bigg(x - \frac{1}{2}\bigg)^2 + (y + 8)^2 = \frac{65}{4}$

Therefore, the equation of the circle is $\bigg(x - \frac{1}{2}\bigg)^2 + (y + 8)^2 = \frac{65}{4}$ .

This section culminates in the grand finale: constructing the equation of the circle using the center and radius values we've meticulously calculated. We start by restating the standard equation of a circle, $(x - h)^2 + (y - k)^2 = r^2$ , reinforcing its importance. Then, we clearly show the substitution of the center coordinates $h = \frac{1}{2}$ and $k = -8$ , along with the radius $r = \frac{\sqrt{65}}{2}$ , into the equation. Each substitution is presented explicitly, ensuring no step is missed. The resulting equation, $\bigg(x - \frac{1}{2}\bigg)^2 + (y + 8)^2 = \frac{65}{4}$ , is the final answer to our problem. The clarity of this section ensures that readers can easily follow the final steps and understand how the equation is derived from the calculated values. This concludes our step-by-step solution, and we have successfully found the equation of the circle given the endpoints of its diameter.

Conclusion

In this article, we have successfully demonstrated how to find the equation of a circle when given the endpoints of its diameter. We followed a step-by-step approach:

Finding the Center: Using the midpoint formula.
Finding the Radius: Using the distance formula.
Writing the Equation: Substituting the center and radius values into the standard equation of a circle.

This method provides a clear and structured way to solve similar problems. Understanding the concepts and applying the formulas correctly are key to mastering these types of questions in geometry. By following these steps, you can confidently determine the equation of a circle given its diameter's endpoints. This article provides a concise recap of the entire process we've undertaken to find the equation of a circle. We reiterate the three key steps: finding the center using the midpoint formula, determining the radius using the distance formula, and constructing the equation by substituting these values into the standard circle equation. This summary reinforces the logical flow of the solution and helps readers consolidate their understanding. The article emphasizes the importance of a structured approach in problem-solving, highlighting how breaking down a complex problem into smaller, manageable steps can lead to a clear and accurate solution. By stressing the correct application of formulas and a solid understanding of the underlying concepts, we empower readers to tackle similar geometry problems with confidence. The concluding statement serves as a final reassurance, encouraging readers to apply the learned techniques to future challenges. This comprehensive guide equips students and math enthusiasts with the tools and knowledge to confidently solve problems involving circles and their equations. Let's continue exploring the fascinating world of geometry!