Finding The Radius Of A Circle With Equation X² + Y² + 8x - 6y + 21 = 0
In the realm of geometry, circles hold a position of fundamental importance. Understanding their properties and equations is crucial for various mathematical applications. One key characteristic of a circle is its radius, the distance from the center to any point on the circumference. This article delves into the process of determining the radius of a circle given its equation in the general form. We will specifically focus on the equation x² + y² + 8x - 6y + 21 = 0, providing a step-by-step approach to solve for the radius and solidify your understanding of circle equations. This exploration will not only equip you with the ability to tackle similar problems but also enhance your overall comprehension of geometric principles. Grasping these concepts is essential for students, educators, and anyone with an interest in mathematics, as circles appear in numerous contexts, from simple diagrams to complex engineering designs. Let's embark on this mathematical journey together, unraveling the mystery of the circle's radius and expanding our knowledge of geometric forms. This detailed explanation aims to clarify every step, ensuring that you not only arrive at the correct answer but also grasp the underlying concepts. By the end of this guide, you will be well-equipped to handle similar problems and appreciate the elegance of mathematical solutions.
Decoding the Circle Equation: Transforming to Standard Form
To find the radius, we must first transform the given equation, x² + y² + 8x - 6y + 21 = 0, into the standard form of a circle's equation. The standard form is expressed as (x - h)² + (y - k)² = r², where (h, k) represents the center of the circle and r denotes the radius. Transforming the equation involves a technique called completing the square. This method allows us to rewrite quadratic expressions in a form that reveals the center and radius of the circle more clearly. The process of completing the square is a cornerstone of algebraic manipulation, enabling us to solve quadratic equations and analyze conic sections effectively. By mastering this technique, you will gain a deeper understanding of how equations represent geometric shapes and how to extract crucial information from them. This section will meticulously guide you through the steps of completing the square, ensuring that you grasp each concept fully. We will break down the process into manageable steps, providing clear explanations and justifications for each manipulation. This will empower you to confidently tackle similar problems and appreciate the power of algebraic transformations in geometric contexts. Understanding the standard form and the process of converting to it is not just about solving this specific problem; it's about building a foundational understanding that will serve you well in more advanced mathematical studies.
Step 1: Grouping Like Terms
The initial step in unraveling the equation is to group the x-terms and y-terms together. This reorganization helps us visualize the components that need to be transformed into perfect squares. By grouping the terms, we set the stage for the subsequent steps in completing the square. This seemingly simple rearrangement is crucial for clarity and organization, making the process more manageable. The equation x² + y² + 8x - 6y + 21 = 0 becomes (x² + 8x) + (y² - 6y) + 21 = 0. This step might seem straightforward, but it is essential for setting up the problem correctly. Without this initial grouping, the subsequent steps of completing the square would be significantly more challenging. This rearrangement is a testament to the power of organization in mathematical problem-solving. By carefully structuring the equation, we pave the way for a smoother and more intuitive solution. This first step is a crucial foundation upon which the rest of the solution will be built, emphasizing the importance of a methodical approach in mathematics.
Step 2: Completing the Square for x
Now, we focus on the x-terms: x² + 8x. To complete the square, we take half of the coefficient of the x term (which is 8), square it, and add it to the expression. Half of 8 is 4, and 4 squared is 16. So, we add 16 to the x-terms: (x² + 8x + 16). This addition transforms the expression into a perfect square trinomial, which can be factored as (x + 4)². Completing the square is a powerful technique that allows us to rewrite quadratic expressions in a more manageable form. It's a fundamental skill in algebra and is used extensively in various mathematical contexts. This step is crucial for transforming the equation into the standard form of a circle's equation. The addition of 16 allows us to rewrite the x-terms as a squared expression, which is a key component of the standard form. Understanding the logic behind completing the square is essential for mastering this technique. It's not just about memorizing the steps; it's about understanding why we add a specific value to create a perfect square. This step demonstrates the elegance of algebraic manipulation and its ability to simplify complex expressions. By completing the square, we are essentially creating a perfect square trinomial that can be easily factored, leading us closer to the solution.
Step 3: Completing the Square for y
Next, we apply the same process to the y-terms: y² - 6y. Half of the coefficient of the y term (which is -6) is -3, and -3 squared is 9. Therefore, we add 9 to the y-terms: (y² - 6y + 9). This creates another perfect square trinomial, which factors as (y - 3)². Just like with the x-terms, completing the square for the y-terms is a crucial step in transforming the equation into standard form. This process reinforces the concept of creating perfect square trinomials, which are essential for solving various algebraic problems. The symmetry in the process for x and y highlights the underlying mathematical principles at play. Understanding this symmetry can make the process of completing the square more intuitive. This step further demonstrates the power of algebraic manipulation in simplifying complex expressions. By completing the square for both x and y, we are well on our way to expressing the equation in its standard form, which will reveal the center and radius of the circle. This methodical approach ensures that we handle each component of the equation with precision, leading to a clear and accurate solution.
Step 4: Adjusting the Equation and Isolating the Constant
Since we added 16 and 9 to the left side of the equation, we must also add them to the right side to maintain the equality. This gives us: (x² + 8x + 16) + (y² - 6y + 9) + 21 = 16 + 9. Now, we rewrite the perfect square trinomials and simplify: (x + 4)² + (y - 3)² + 21 = 25. Next, we isolate the constant term by subtracting 21 from both sides: (x + 4)² + (y - 3)² = 25 - 21, which simplifies to (x + 4)² + (y - 3)² = 4. This step is crucial for ensuring that the equation remains balanced and that we are working with an equivalent expression. The act of adding the same values to both sides is a fundamental principle of algebraic manipulation. This step also highlights the importance of maintaining accuracy throughout the problem-solving process. By carefully tracking the changes made to the equation, we ensure that the final result is correct. The isolation of the constant term is the final piece of the puzzle in transforming the equation into standard form. This step sets us up perfectly to identify the radius of the circle. The clarity and precision of this step demonstrate the importance of a methodical approach in mathematics.
Determining the Radius: Unveiling the Circle's Size
Now that we have the equation in the standard form, (x + 4)² + (y - 3)² = 4, we can easily determine the radius. Recall that the standard form is (x - h)² + (y - k)² = r², where r² represents the square of the radius. In our equation, r² = 4. To find the radius, we take the square root of both sides: r = √4. Therefore, the radius of the circle is 2 units. This step is the culmination of our efforts, where the algebraic manipulations we performed earlier pay off by revealing the radius of the circle. The ease with which we can now identify the radius underscores the power of transforming the equation into standard form. This process highlights the connection between algebraic expressions and geometric properties. The radius, a fundamental characteristic of the circle, is directly extracted from the standard form of the equation. This step demonstrates the elegance and efficiency of mathematical methods. By understanding the relationship between the equation and the geometric shape, we can quickly and accurately determine key properties. The solution, r = 2 units, is the final answer to our problem, but the journey we took to arrive at this answer is just as important. The process of completing the square and transforming the equation provides valuable insights into algebraic manipulation and geometric interpretation.
Conclusion: The Radius Revealed and Mathematical Mastery Achieved
In conclusion, by meticulously applying the technique of completing the square, we successfully transformed the equation x² + y² + 8x - 6y + 21 = 0 into the standard form (x + 4)² + (y - 3)² = 4. This transformation allowed us to readily identify the radius of the circle as 2 units. This exercise not only provides the answer to the specific question but also reinforces fundamental mathematical concepts and problem-solving skills. The ability to manipulate equations and connect them to geometric representations is a crucial skill in mathematics and related fields. The process of completing the square is a versatile technique that can be applied to various problems, making it an invaluable tool in your mathematical arsenal. By mastering these techniques, you gain a deeper understanding of the relationships between algebraic expressions and geometric shapes. This understanding empowers you to tackle more complex problems with confidence and clarity. The journey of solving this problem is just as important as the answer itself. The steps we took, the techniques we applied, and the insights we gained all contribute to our mathematical growth. This detailed explanation aims to provide not only a solution but also a comprehensive understanding of the underlying principles. By grasping these concepts, you are well-equipped to tackle future challenges and appreciate the beauty and elegance of mathematics.
Therefore, the correct answer is A. 2 units.
