Finding The Radius Of A Circle $x^2 + Y^2 - 10x + 6y + 18 = 0$ A Step-by-Step Guide

Determining the radius of a circle from its equation is a fundamental concept in coordinate geometry. This article delves into the process of finding the radius of a circle given its equation in the general form. We will specifically focus on the equation $x^2 + y^2 - 10x + 6y + 18 = 0$, illustrating a step-by-step method to arrive at the solution. By the end of this guide, you'll not only be able to solve this particular problem but also grasp the underlying principles for tackling similar challenges. Our journey will begin by revisiting the standard equation of a circle, followed by the technique of completing the square, and finally, extracting the radius from the transformed equation.

The Standard Equation of a Circle: A Foundation for Understanding

At the heart of our exploration lies the standard equation of a circle, a cornerstone concept in analytic geometry. This equation provides a concise and powerful way to represent a circle's properties within a Cartesian coordinate system. Understanding this standard form is crucial for unraveling the characteristics of any circle, including its center and, most importantly for our discussion, its radius. The standard equation is expressed as:

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

Where:

• (h, k) represents the coordinates of the circle's center.
• r denotes the radius of the circle, which is the distance from the center to any point on the circle's circumference.

This elegant equation encapsulates the very essence of a circle: a set of points equidistant (the radius) from a central point. The beauty of this form lies in its ability to immediately reveal the circle's center and radius, making it an invaluable tool for geometric analysis. When we encounter a circle's equation in a different form, such as the general form, our goal is often to manipulate it into this standard form. By doing so, we unlock the circle's hidden properties and gain a clear understanding of its position and size within the coordinate plane. In the context of our problem, where we are given the equation $x^2 + y^2 - 10x + 6y + 18 = 0$, our primary objective is to transform this equation into the standard form. This transformation will allow us to directly identify the radius, which is the key to solving the problem. The process of transforming the equation involves a technique called completing the square, which we will explore in detail in the next section. Before we move on, it's essential to appreciate the significance of the standard equation. It's not merely a formula; it's a visual representation of a circle's fundamental properties. Each term in the equation corresponds to a specific geometric aspect of the circle, making it a powerful tool for both understanding and manipulating circular shapes in the coordinate plane. The ability to move seamlessly between the general form and the standard form is a crucial skill in analytic geometry, and it forms the basis for solving a wide range of problems related to circles. As we proceed with our exploration, keep the standard equation in mind as our ultimate destination. It's the key that will unlock the radius of the circle defined by the equation $x^2 + y^2 - 10x + 6y + 18 = 0$.

Completing the Square: The Key to Transformation

To convert the given equation, $x^2 + y^2 - 10x + 6y + 18 = 0$, into the standard form, we employ a powerful algebraic technique known as completing the square. This method allows us to rewrite quadratic expressions as perfect squares, which is essential for revealing the circle's center and radius. Completing the square involves manipulating the equation by adding and subtracting specific constants to create perfect square trinomials for both the x and y terms. Let's break down the process step-by-step:

1. Group the x and y terms: Begin by rearranging the equation, grouping the x terms together and the y terms together, and moving the constant term to the right side of the equation:

   $(x^2 - 10x) + (y^2 + 6y) = -18$

2. Complete the square for x: To complete the square for the x terms, take half of the coefficient of the x term (-10), square it ((-5)^2 = 25), and add it to both sides of the equation:

   $(x^2 - 10x + 25) + (y^2 + 6y) = -18 + 25$

3. Complete the square for y: Similarly, for the y terms, take half of the coefficient of the y term (6), square it ((3)^2 = 9), and add it to both sides of the equation:

   $(x^2 - 10x + 25) + (y^2 + 6y + 9) = -18 + 25 + 9$

4. Rewrite as squared terms: Now, rewrite the expressions in parentheses as perfect squares:

   $(x - 5)^2 + (y + 3)^2 = 16$

This transformed equation is now in the standard form of a circle's equation. By completing the square, we have successfully converted the general form into the standard form, making it easy to identify the circle's center and radius. The key to mastering completing the square lies in understanding the relationship between the coefficients of the quadratic and linear terms and the constant term needed to create a perfect square trinomial. This technique is not only useful for circles but also for other conic sections like ellipses and hyperbolas. The ability to complete the square is a fundamental skill in algebra and analytic geometry, providing a powerful tool for solving a wide range of problems. In our case, it has allowed us to transform the given equation into a form that directly reveals the circle's radius, which is the next step in our solution. Before we move on, it's worth noting that completing the square is a systematic process that can be applied to any quadratic expression. By following the steps outlined above, you can confidently transform any equation into a form that reveals its underlying properties. The importance of this technique cannot be overstated, as it forms the basis for solving many problems in mathematics and physics. As we proceed to the next section, we will see how the standard form of the equation directly leads us to the solution for the radius of the circle.

Extracting the Radius: The Final Step

Now that we have successfully transformed the equation $x^2 + y^2 - 10x + 6y + 18 = 0$ into its standard form, $(x - 5)^2 + (y + 3)^2 = 16$, the final step is to extract the radius. Recall that the standard equation of a circle is given by:

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

Where:

• (h, k) is the center of the circle.
• r is the radius of the circle.

By comparing our transformed equation, $(x - 5)^2 + (y + 3)^2 = 16$, with the standard form, we can easily identify the values of h, k, and r^2. In this case, we have:

• h = 5
• k = -3
• $r^2$ = 16

Therefore, the center of the circle is (5, -3). However, our primary goal is to find the radius, r. To do this, we simply take the square root of $r^2$:

r = √16 = 4

Thus, the radius of the circle is 4 units. This completes our solution. We have successfully determined the radius of the circle defined by the equation $x^2 + y^2 - 10x + 6y + 18 = 0$. The process involved transforming the equation into standard form using the technique of completing the square and then extracting the radius by taking the square root of the constant term on the right side of the equation. This final step highlights the power of the standard equation of a circle. Once the equation is in this form, the radius is readily apparent, making it a straightforward task to determine the circle's size. The ability to extract the radius from the standard equation is a crucial skill in analytic geometry, allowing us to quickly and easily analyze the properties of circles. In summary, we have demonstrated a complete method for finding the radius of a circle given its equation in the general form. This method involves a combination of algebraic manipulation and a clear understanding of the standard equation of a circle. By mastering these techniques, you can confidently tackle similar problems and gain a deeper appreciation for the beauty and power of coordinate geometry. Before we conclude, it's important to emphasize the importance of accuracy in each step of the process. A small error in completing the square or extracting the square root can lead to an incorrect answer. Therefore, it's crucial to double-check your work and ensure that each step is performed correctly. With practice and attention to detail, you can become proficient in finding the radius of a circle from its equation.

Conclusion: Mastering the Circle's Equation

In conclusion, we have successfully navigated the process of finding the radius of a circle whose equation is given as $x^2 + y^2 - 10x + 6y + 18 = 0$. Our journey began with understanding the standard equation of a circle, a fundamental concept that provides a clear representation of a circle's center and radius. We then delved into the technique of completing the square, a powerful algebraic tool that allowed us to transform the given equation into the standard form. This transformation was the key to unlocking the circle's properties. Finally, we demonstrated how to extract the radius from the standard equation, arriving at the solution of 4 units. This comprehensive approach highlights the interconnectedness of different concepts in mathematics. The standard equation of a circle provides a framework for understanding circles in the coordinate plane, while completing the square offers a method for manipulating equations to reveal their underlying structure. Extracting the radius is the final step in this process, allowing us to quantify the size of the circle. The ability to move seamlessly between these concepts is a hallmark of mathematical proficiency. By mastering these techniques, you can confidently tackle a wide range of problems involving circles and other conic sections. This understanding extends beyond the realm of mathematics, finding applications in various fields such as physics, engineering, and computer graphics. The circle, a fundamental geometric shape, plays a crucial role in many natural and man-made phenomena. From the orbits of planets to the design of wheels, circles are ubiquitous in our world. Therefore, a solid understanding of circles and their properties is essential for anyone pursuing a career in science or technology. As we conclude this exploration, it's important to emphasize the value of practice. The more you work with circles and their equations, the more comfortable and confident you will become in applying these techniques. Challenge yourself with different equations and explore the various ways in which circles can be represented and manipulated. With dedication and perseverance, you can master the circle's equation and unlock its secrets. The journey of mathematical discovery is a rewarding one, and the knowledge and skills you gain along the way will serve you well in your future endeavors.

The correct answer is B. 4 units.