Finding The Radius Of A Circle Given Its Equation

JU07/10/2025 08, 2025 by THE IDEN 50 views

Unveiling the Radius of a Circle The radius of a circle is a fundamental property that defines its size. When given the equation of a circle in a general form, extracting the radius requires a systematic approach. In this article, we will delve into the process of determining the radius of a circle from its equation, focusing on the specific example of *x*<sup>2</sup> + *y*<sup>2</sup> - 2*x* + 8*y* - 47 = 0. Understanding this process is crucial for various applications in geometry, physics, and engineering, where circles and their properties play a significant role.

Understanding the General Equation of a Circle

Before we dive into the specifics of our given equation, it's essential to grasp the general form of a circle's equation. The general equation of a circle is expressed as: x² + y² + 2gx + 2fy + c = 0 Where (-g, -f) represents the center of the circle and the radius (r) is given by the formula: r = √(g² + f² - c) This formula is derived from the standard form of a circle's equation, which we will explore next. The general equation provides a flexible way to represent circles, but it requires manipulation to extract key information such as the center and radius. This is where the technique of completing the square comes into play, allowing us to transform the general equation into the standard form. Understanding the relationship between the general form and the standard form is crucial for solving problems involving circles.

Transforming to the Standard Form

To find the radius, we need to transform the given equation into the standard form of a circle's equation, which is: (x - h)² + (y - k)² = r² Where (h, k) is the center of the circle and r is the radius. This transformation is achieved through a process called completing the square. Completing the square involves rearranging the terms in the equation and adding constants to both sides to create perfect square trinomials for both the x and y terms. This technique allows us to rewrite the equation in a form that directly reveals the center and radius of the circle. Let's apply this technique to our equation: x² + y² - 2x + 8y - 47 = 0 First, we group the x terms and the y terms together: (x² - 2x) + (y² + 8y) = 47 Next, we complete the square for the x terms. To do this, we take half of the coefficient of the x term (-2), square it ((-1)² = 1), and add it to both sides of the equation. We do the same for the y terms, taking half of the coefficient of the y term (8), squaring it ((4)² = 16), and adding it to both sides. This process ensures that we maintain the equality of the equation while creating perfect square trinomials. By carefully applying the steps of completing the square, we can transform the general equation into the standard form, making it easy to identify the center and radius of the circle.

Completing the Square: A Step-by-Step Guide

Let's complete the square for the x terms: x² - 2x. To complete the square, we need to add and subtract (b/2)², where b is the coefficient of the x term. In this case, b = -2, so (b/2)² = (-2/2)² = 1. Thus, we add and subtract 1: (x² - 2x + 1) - 1. The expression inside the parentheses is now a perfect square trinomial: (x - 1)² - 1. Now, let's complete the square for the y terms: y² + 8y. Here, b = 8, so (b/2)² = (8/2)² = 16. We add and subtract 16: (y² + 8y + 16) - 16. Again, we have a perfect square trinomial: (y + 4)² - 16. By systematically applying the completing the square technique, we transform the quadratic expressions into perfect square trinomials, which are essential for rewriting the equation in standard form. This meticulous process ensures accuracy and clarity in determining the circle's properties.

Rewriting the Equation

Now, substitute these back into the equation: (x - 1)² - 1 + (y + 4)² - 16 = 47 Combine the constants: (x - 1)² + (y + 4)² = 47 + 1 + 16 (x - 1)² + (y + 4)² = 64 This is now in the standard form of a circle's equation, (x - h)² + (y - k)² = r², where the center is (h, k) and the radius is r. The transformation to standard form is a crucial step, as it allows us to directly read off the center and radius of the circle. The process involves careful algebraic manipulation and a clear understanding of the relationship between the general and standard forms of the equation. By rewriting the equation, we've unveiled the circle's fundamental properties, setting the stage for determining its radius.

Identifying the Radius

From the standard form, (x - 1)² + (y + 4)² = 64, we can see that r² = 64. To find the radius r, we take the square root of both sides: r = √64 r = 8 Therefore, the radius of the circle is 8 units. Identifying the radius from the standard form is a straightforward process, involving taking the square root of the constant term on the right-hand side of the equation. This direct relationship between the standard form and the circle's properties highlights the importance of mastering the transformation techniques. The radius, in this case, is a whole number, indicating a clear and well-defined circle in the coordinate plane. The radius, being a fundamental property, helps us understand the size and scale of the circle, which is crucial in various applications.

Center of the Circle

While we focused on finding the radius, it's worth noting that the standard form also reveals the center of the circle. In the equation (x - 1)² + (y + 4)² = 64, the center is (h, k) = (1, -4). The center and radius together completely define a circle's position and size in the coordinate plane. The center serves as the focal point around which the circle is drawn, and its coordinates provide valuable information about the circle's location relative to the axes. Understanding both the center and radius is essential for a comprehensive understanding of the circle's geometry and its relationship to other geometric figures.

Conclusion

In conclusion, by transforming the given equation x² + y² - 2x + 8y - 47 = 0 into the standard form (x - 1)² + (y + 4)² = 64, we successfully determined that the radius of the circle is 8 units. This process involved completing the square, a powerful technique for manipulating quadratic equations. Understanding how to find the radius of a circle from its equation is a valuable skill in mathematics and its applications. The ability to transform equations and extract key information is a hallmark of mathematical proficiency. By mastering these techniques, we gain a deeper understanding of geometric shapes and their properties, paving the way for more advanced problem-solving in various fields.

This method not only allows us to find the radius but also the center of the circle, providing a complete description of the circle's properties. The combination of algebraic manipulation and geometric interpretation is a key aspect of mathematical thinking. The radius, being a fundamental property, helps us understand the size and scale of the circle, which is crucial in various applications. The ability to determine the radius from the equation is a powerful tool in geometry and beyond, enabling us to solve problems involving circles in diverse contexts.