Finding The Radius Of A Circle Given Its Equation $4x^2 + 4y^2 - 24x - 32y + 72 = 0$

In this article, we'll explore how to determine the radius of a circle given its equation in the general form. Specifically, we will focus on the equation $4x^2 + 4y^2 - 24x - 32y + 72 = 0$. Understanding the properties of circles and their equations is fundamental in analytic geometry, and this exercise provides a practical application of those principles. Let's embark on this mathematical journey together!

Understanding the General Equation of a Circle

Before we dive into the specifics of our given equation, it's crucial to understand the general form of a circle's equation. The general form is given by: $Ax^2 + Ay^2 + Bx + Cy + D = 0$, where A, B, C, and D are constants. However, the standard form of a circle's equation is much more informative when it comes to quickly identifying the circle's center and radius. The standard form is represented as: $(x - h)^2 + (y - k)^2 = r^2$, where (h, k) represents the center of the circle and r represents the radius. Converting the general form to the standard form is the key to solving our problem. The process involves completing the square, a powerful algebraic technique that allows us to rewrite quadratic expressions in a more manageable form. Understanding this conversion is not just a mathematical exercise; it's a tool that finds applications in various fields, including physics, engineering, and computer graphics. For instance, in physics, understanding circular motion often requires manipulating equations of circles. In engineering, designing circular structures or components relies on a solid grasp of circular geometry. And in computer graphics, circles are fundamental building blocks for creating various shapes and images. Thus, mastering the equation of a circle is a valuable skill with far-reaching implications.

Transforming the Equation: Completing the Square

Our main task is to transform the given equation, $4x^2 + 4y^2 - 24x - 32y + 72 = 0$, into the standard form. The first step is to simplify the equation by dividing all terms by the common coefficient of the squared terms, which is 4 in this case. This gives us: $x^2 + y^2 - 6x - 8y + 18 = 0$. Now, we need to group the x terms and the y terms together and move the constant term to the right side of the equation: $(x^2 - 6x) + (y^2 - 8y) = -18$. The next step is the heart of the process: completing the square for both the x and y terms. To complete the square for the x terms, we take half of the coefficient of the x term (-6), square it ((-3)^2 = 9), and add it to both sides of the equation. We do the same for the y terms: take half of the coefficient of the y term (-8), square it ((-4)^2 = 16), and add it to both sides. This gives us: $(x^2 - 6x + 9) + (y^2 - 8y + 16) = -18 + 9 + 16$. Now, we can rewrite the expressions in parentheses as perfect squares: $(x - 3)^2 + (y - 4)^2 = 7$. This equation is now in the standard form of a circle's equation, making it easy to identify the center and the radius. The center of the circle is (3, 4), and the square of the radius is 7. Therefore, the radius is the square root of 7. This process of completing the square is a fundamental technique in algebra and calculus. It's used not only for circles but also for other conic sections like parabolas, ellipses, and hyperbolas. Mastering this technique unlocks a deeper understanding of these geometric shapes and their properties.

Identifying the Radius

From the transformed equation, $(x - 3)^2 + (y - 4)^2 = 7$, we can directly read off the radius. Recall that the standard form of a circle's equation is $(x - h)^2 + (y - k)^2 = r^2$, where r is the radius. Comparing our equation with the standard form, we see that $r^2 = 7$. Therefore, the radius, r, is the square root of 7, which we write as $r = \sqrt{7}$. The radius is a fundamental property of a circle, defining its size and influencing many of its other characteristics. For example, the circumference of a circle is given by $2\pi r$, and the area is given by $\pi r^2$. Knowing the radius allows us to calculate these important measures. Moreover, the radius plays a crucial role in various applications of circles. In engineering, the radius is a key parameter in designing circular components like gears, wheels, and pipes. In architecture, circular shapes are often used for aesthetic and structural purposes, and the radius is essential for determining the dimensions of these structures. In physics, the radius is a critical factor in understanding circular motion, such as the orbits of planets or the rotation of objects. Therefore, accurately determining the radius of a circle from its equation is a valuable skill with practical applications in many fields.

The Significance of the Radius in Geometry and Beyond

The radius of a circle, as we've seen, is not just a number; it's a fundamental characteristic that governs many of a circle's properties. Understanding the radius allows us to delve deeper into the geometry of circles and their relationship to other geometric shapes. For instance, the diameter of a circle, which is twice the radius, is the longest line segment that can be drawn within the circle. The circumference, which is the distance around the circle, is directly proportional to the radius. Similarly, the area enclosed by the circle is proportional to the square of the radius. But the significance of the radius extends far beyond basic geometry. In trigonometry, the unit circle (a circle with a radius of 1) is a cornerstone for understanding trigonometric functions like sine, cosine, and tangent. These functions are essential for modeling periodic phenomena in physics, engineering, and other fields. In calculus, the concept of the radius of curvature is used to describe the curvature of a curve at a particular point. This is a crucial concept in understanding the motion of objects along curved paths. Moreover, the radius plays a vital role in various applications. In optics, the radius of curvature of lenses is a critical parameter in determining their focusing properties. In telecommunications, the radius of a circular waveguide affects the propagation of electromagnetic waves. And in astronomy, the radii of celestial bodies are fundamental data points for understanding their size, mass, and other properties. Thus, mastering the concept of the radius and its determination is not just an academic exercise; it's a key to unlocking a deeper understanding of the world around us.

Therefore, the radius of the circle defined by the equation $4x^2 + 4y^2 - 24x - 32y + 72 = 0$ is $\sqrt{7}$. This exercise highlights the importance of understanding the relationship between the general and standard forms of a circle's equation and the power of completing the square as a problem-solving technique. This foundational knowledge is invaluable for further exploration in mathematics and its applications.