Finding The Equation Of A Circle Given Diameter Endpoints A Step By Step Guide

In the fascinating world of geometry, circles hold a special place. Their perfect symmetry and elegant equations have captivated mathematicians for centuries. One common problem encountered in coordinate geometry 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 clear and concise method for finding the circle's equation. We will explore the underlying principles, the formulas involved, and illustrate the process with a detailed example. Mastering this skill is crucial for anyone studying analytical geometry, as it combines concepts of distance, midpoint, and the standard equation of a circle.

Understanding the Circle's Equation

Before diving into the solution, let's first understand the standard equation of a circle. A circle with center $(h, k)$ and radius $r$ can be represented by the equation:

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

Here, $(x, y)$ represents any point on the circle's circumference. The center $(h, k)$ is the pivotal point around which the circle is drawn, and the radius $r$ is the distance from the center to any point on the circle. This equation is derived from the Pythagorean theorem and the distance formula, which are fundamental concepts in coordinate geometry. Understanding this equation is key to solving problems related to circles, as it provides a direct link between the circle's geometric properties and its algebraic representation. The equation allows us to describe the circle's position and size in the coordinate plane, making it a powerful tool in mathematical analysis and problem-solving.

Steps to Find the Circle's Equation

To find the equation of a circle given the endpoints of a diameter, we follow a systematic approach that involves finding the center and the radius of the circle. The diameter, by definition, passes through the center of the circle, and its endpoints provide crucial information for determining these parameters. The process typically involves the following steps:

1. Find the Center: The center of the circle is the midpoint of the diameter. Given the endpoints $J(x_1, y_1)$ and $K(x_2, y_2)$, the midpoint (center) $(h, k)$ can be found using the midpoint formula:

   $$h = \frac{x_1 + x_2}{2}$$

   $$k = \frac{y_1 + y_2}{2}$$

   This formula is derived from the concept of averaging the coordinates of the endpoints, providing a straightforward method to locate the center of the circle. The midpoint formula is a fundamental tool in coordinate geometry, and its application here simplifies the process of finding the circle's center.

2. Find the Radius: The radius of the circle is half the length of the diameter. The length of the diameter can be found using the distance formula between the two endpoints $J(x_1, y_1)$ and $K(x_2, y_2)$:

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

   The radius $r$ is then half of this distance:

   $$r = \frac{d}{2}$$

   Alternatively, the radius can be found by calculating the distance between the center $(h, k)$ and one of the endpoints using the distance formula. This approach is equally valid and can sometimes be more convenient depending on the specific problem.

3. Write the Equation: Once the center $(h, k)$ and the radius $r$ are known, the equation of the circle can be written in the standard form:

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

   This step involves substituting the values of $h$, $k$, and $r$ into the standard equation, resulting in the specific equation that represents the circle in question. The equation provides a complete description of the circle's properties, allowing us to analyze and manipulate it further.

Detailed Example

Let's consider the example provided: The points $J(-8, 9)$ and $K(-2, -5)$ are endpoints of a diameter of circle $C$. We will follow the steps outlined above to find the equation of circle $C$.

1. Find the Center

Using the midpoint formula, we find the center $(h, k)$:

$$h = \frac{-8 + (-2)}{2} = \frac{-10}{2} = -5$$

$$k = \frac{9 + (-5)}{2} = \frac{4}{2} = 2$$

Thus, the center of the circle is $(-5, 2)$. This means the circle is centered at the point where the x-coordinate is -5 and the y-coordinate is 2. This location is crucial for defining the circle's position in the coordinate plane.

2. Find the Radius

First, we find the length of the diameter using the distance formula:

$$d = \sqrt{((-2) - (-8))^2 + ((-5) - 9)^2}$$

$$d = \sqrt{(6)^2 + (-14)^2}$$

$$d = \sqrt{36 + 196}$$

$$d = \sqrt{232}$$

Now, we find the radius by dividing the diameter by 2:

$$r = \frac{\sqrt{232}}{2}$$

Alternatively, we can find the radius by calculating the distance between the center $(-5, 2)$ and one of the endpoints, say $J(-8, 9)$:

$$r = \sqrt{((-8) - (-5))^2 + (9 - 2)^2}$$

$$r = \sqrt{(-3)^2 + (7)^2}$$

$$r = \sqrt{9 + 49}$$

$$r = \sqrt{58}$$

Therefore, $r^2 = 58$. This value is essential for completing the circle's equation. The radius, whether calculated directly or through the diameter, provides the necessary scale for the circle.

3. Write the Equation

Now that we have the center $(-5, 2)$ and $r^2 = 58$, we can write the equation of the circle:

$$(x - (-5))^2 + (y - 2)^2 = 58$$

$$(x + 5)^2 + (y - 2)^2 = 58$$

This equation represents the circle $C$ in the coordinate plane. It precisely defines the set of all points that lie on the circumference of the circle. The equation is the culmination of our calculations, providing a concise and complete description of the circle.

Analyzing the Options

Comparing our result with the given options:

A. $(x-5)^2+(y+2)^2=58$

B. $(x-5)^2+(y+2)^2=232$

C. $(x+5)^2+(y-2)^2=58$

D. $(x+5)^2+(y-2)^2=232$

We can see that option C, $(x+5)^2+(y-2)^2=58$, matches our derived equation. Therefore, the correct answer is C.

Key Takeaways

This problem demonstrates a fundamental concept in coordinate geometry: finding the equation of a circle given the endpoints of its diameter. The key steps involve:

• Calculating the center of the circle using the midpoint formula.
• Determining the radius using the distance formula (either from the diameter or from the center to an endpoint).
• Substituting the center and radius into the standard equation of a circle.

Understanding and applying these steps allows for the efficient solution of such problems. The ability to manipulate these equations and formulas is crucial for success in analytical geometry and related fields.

Tips for Solving Circle Equation Problems

When dealing with circle equation problems, consider the following tips to enhance your problem-solving skills:

• Visualize the Circle: Sketching a rough diagram of the circle and the given points can help in understanding the problem better. This visual representation can provide insights into the relationships between the center, radius, and endpoints.
• Double-Check Calculations: Errors in calculations can lead to incorrect answers. Always double-check your calculations, especially when using the distance and midpoint formulas. Accuracy is paramount in these types of problems.
• Understand the Formulas: Having a strong understanding of the midpoint and distance formulas, as well as the standard equation of a circle, is essential. Memorizing the formulas is not enough; understanding their derivation and application is crucial.
• Simplify Radicals: If the radius involves a radical, simplify it as much as possible. This will make it easier to compare your answer with the given options.
• Practice Regularly: Like any mathematical skill, solving circle equation problems requires practice. Solve a variety of problems to build your confidence and competence.

By mastering these techniques and tips, you can confidently tackle any problem involving the equation of a circle.

Common Mistakes to Avoid

While solving circle equation problems, there are some common mistakes that students often make. Being aware of these pitfalls can help you avoid them and improve your accuracy:

• Incorrectly Applying the Midpoint Formula: Ensure that you are adding the x-coordinates and y-coordinates separately before dividing by 2. Mixing up the coordinates can lead to an incorrect center.
• Forgetting to Square the Radius: The standard equation of a circle uses $r^2$, not $r$. Make sure to square the radius when writing the equation. This is a common oversight that can easily be avoided.
• Confusing the Signs: Pay close attention to the signs in the standard equation $(x - h)^2 + (y - k)^2 = r^2$. The coordinates of the center are subtracted from $x$ and $y$, so a center of $(-5, 2)$ corresponds to $(x + 5)$ and $(y - 2)$ in the equation.
• Using the Diameter Instead of the Radius: Remember that the radius is half the length of the diameter. Using the diameter as the radius will result in an incorrect equation.
• Misinterpreting the Question: Always read the question carefully to ensure you understand what is being asked. Sometimes, the question may ask for the diameter instead of the radius, or vice versa.

By being mindful of these common mistakes, you can significantly reduce the likelihood of errors and improve your problem-solving accuracy.

Conclusion

Finding the equation of a circle given the endpoints of a diameter is a fundamental skill in coordinate geometry. By following a systematic approach, utilizing the midpoint and distance formulas, and understanding the standard equation of a circle, you can confidently solve these types of problems. Remember to practice regularly, double-check your calculations, and be aware of common mistakes to ensure accuracy. With these tools at your disposal, you'll be well-equipped to navigate the world of circles in mathematics.

This detailed explanation provides a comprehensive guide to finding the equation of a circle, ensuring clarity and understanding for anyone studying this topic. The structured approach, detailed example, and helpful tips make this article a valuable resource for mastering circle equations.

Original Question

The points $J(-8,9)$ and $K(-2,-5)$ are endpoints of a diameter of circle $C$. Which equation represents circle $C$?

A. $(x-5)^2+(y+2)^2=58$

B. $(x-5)^2+(y+2)^2=232$

C. $(x+5)^2+(y-2)^2=58$

D. $(x+5)^2+(y-2)^2=232$

Revised Question: Determining Circle Equation from Diameter Endpoints

If the endpoints of a diameter of circle $C$ are the points $J(-8,9)$ and $K(-2,-5)$, what is the equation that correctly represents circle $C$? Choose from the following options:

A. $(x-5)^2+(y+2)^2=58$

B. $(x-5)^2+(y+2)^2=232$

C. $(x+5)^2+(y-2)^2=58$

D. $(x+5)^2+(y-2)2=232