Determine The X- And Y-intercepts Of The Circle Given By The Equation 50x^2 + 50y^2 - 20x - 25y - 23 = 0.
In the realm of analytical geometry, circles hold a fundamental position. Their equations, particularly when presented in the general form, might seem daunting at first glance. However, by employing a systematic, step-by-step approach, we can unravel the intricacies of these equations and accurately sketch the circle they represent. This article delves into the process of transforming a general form equation of a circle into its standard form, thereby revealing the circle's center and radius, and subsequently, determining its intercepts. We will use the example equation 50x^2 + 50y^2 - 20x - 25y - 23 = 0 to illustrate each step of the process. Understanding the process of converting general form equations to standard form is not just a mathematical exercise; it's a crucial skill for anyone working with geometric shapes and their applications in fields like physics, engineering, and computer graphics. The ability to quickly and accurately determine the properties of a circle from its equation is invaluable in a variety of problem-solving scenarios. This guide aims to provide a comprehensive understanding of this process, making it accessible to students and professionals alike.
Understanding the General and Standard Forms of a Circle Equation
Before we dive into the step-by-step process, it's crucial to understand the two primary forms of a circle's equation: the general form and the standard form. The general form, typically expressed as Ax^2 + Ay^2 + Dx + Ey + F = 0, where A, D, E, and F are constants, often obscures the circle's essential properties. The example equation, 50x^2 + 50y^2 - 20x - 25y - 23 = 0, perfectly exemplifies this form. The coefficients and constant terms are present, but the center and radius aren't immediately apparent. This is where the standard form comes into play. The standard form, represented as (x - h)^2 + (y - k)^2 = r^2, provides a clear representation of the circle's center (h, k) and its radius r. Converting from the general form to the standard form is the key to unlocking the circle's properties. This transformation allows us to visualize the circle's position on the coordinate plane and its size. The process involves algebraic manipulation, specifically completing the square, which we will explore in detail in the subsequent sections. Understanding these two forms and the relationship between them is the foundation for graphing circles and solving related problems. It's a fundamental concept in analytic geometry that underpins many advanced topics.
Step-by-Step Process: From General Form to Graph
The journey from the general form equation to sketching the circle's graph involves several key steps. These steps systematically transform the equation into a more manageable form, revealing the circle's center and radius. Let's break down each step using our example equation, 50x^2 + 50y^2 - 20x - 25y - 23 = 0. The first step is to rearrange and group the terms. We group the x-terms together, the y-terms together, and move the constant term to the right side of the equation. This sets the stage for completing the square. The next crucial step involves completing the square for both the x and y terms. This is an algebraic technique that transforms a quadratic expression into a perfect square trinomial. By adding and subtracting appropriate constants, we can rewrite the x and y terms in a squared form. Once we've completed the square, we can rewrite the equation in the standard form (x - h)^2 + (y - k)^2 = r^2. This form directly reveals the circle's center (h, k) and its radius r. With the center and radius known, sketching the graph becomes a straightforward process. We plot the center on the coordinate plane and then use the radius to determine the circle's extent in all directions. This provides a visual representation of the circle defined by the original equation. Finally, we can determine the x- and y-intercepts of the circle, which are the points where the circle intersects the x-axis and y-axis, respectively. These intercepts provide additional key points for accurately sketching the graph and understanding the circle's position in the coordinate plane.
(a) Rewriting the Equation and Completing the Square
The initial hurdle in graphing a circle from its general form equation lies in transforming it into the standard form. This process primarily involves rewriting the equation and completing the square. Starting with our equation, 50x^2 + 50y^2 - 20x - 25y - 23 = 0, the first step is to divide the entire equation by the coefficient of the squared terms (50 in this case) to simplify it. This gives us x^2 + y^2 - (2/5)x - (1/2)y - 23/50 = 0. This simplification makes the process of completing the square more manageable. Next, we rearrange the terms, grouping the x-terms and y-terms together and moving the constant term to the right side of the equation: (x^2 - (2/5)x) + (y^2 - (1/2)y) = 23/50. Now, we embark on the process of completing the square for both x and y. For the x-terms, we take half of the coefficient of the x term (-2/5), which is -1/5, square it (1/25), and add it to both sides of the equation. Similarly, for the y-terms, we take half of the coefficient of the y term (-1/2), which is -1/4, square it (1/16), and add it to both sides. This yields: (x^2 - (2/5)x + 1/25) + (y^2 - (1/2)y + 1/16) = 23/50 + 1/25 + 1/16. The expressions within the parentheses are now perfect square trinomials, which can be factored into squared binomials. Completing the square is a pivotal step as it allows us to rewrite the general form equation into the standard form, which directly reveals the circle's center and radius. This algebraic manipulation is a powerful tool in analytic geometry, enabling us to extract key information from seemingly complex equations.
(b) Determining the Center and Radius
With the equation rewritten and the square completed, we arrive at a pivotal point: determining the center and radius of the circle. After completing the square in the previous step, our equation now looks like this: (x - 1/5)^2 + (y - 1/4)^2 = 23/50 + 1/25 + 1/16. To find a common denominator and simplify the right side, we get (x - 1/5)^2 + (y - 1/4)^2 = 184/400 + 16/400 + 25/400, which simplifies further to (x - 1/5)^2 + (y - 1/4)^2 = 225/400. Reducing the fraction, we have (x - 1/5)^2 + (y - 1/4)^2 = 9/16. Now, the equation is in the standard form (x - h)^2 + (y - k)^2 = r^2. By comparing our equation with the standard form, we can directly identify the center and the radius. The center (h, k) is (1/5, 1/4). The radius r is the square root of the right side of the equation, which is √(9/16) = 3/4. Therefore, the circle has a center at (1/5, 1/4) and a radius of 3/4. This information is the key to sketching the graph of the circle. Knowing the center and radius allows us to accurately position the circle on the coordinate plane and determine its size. The ability to extract this information from the standard form equation is a fundamental skill in analytic geometry.
(c) Graphing the Circle
With the center and radius now determined, the next step is to graph the circle. We know the center is at (1/5, 1/4), which is equivalent to (0.2, 0.25), and the radius is 3/4, or 0.75. To graph the circle, we first plot the center point on the coordinate plane. This point serves as the central reference for our circle. From the center, we use the radius to determine the circle's extent. We can measure out the radius distance in four directions: up, down, left, and right from the center. These four points will lie on the circle's circumference. Specifically, we move 0.75 units up, down, left, and right from the center (0.2, 0.25). This gives us four points: (0.2, 1), (0.2, -0.5), (0.95, 0.25), and (-0.55, 0.25). These points help guide us in sketching the circle's curve. Using these guide points, we carefully draw a smooth, continuous curve that represents the circle. The circle should be centered around the center point we plotted earlier, and the curve should be equidistant from the center at all points. The accuracy of the graph depends on the precision with which we plot the center and measure the radius. A well-drawn graph provides a visual representation of the circle defined by the original equation, allowing us to understand its position and size in the coordinate plane. Graphing the circle is a crucial step in visualizing the equation and understanding its geometric properties. It bridges the gap between the algebraic representation and the geometric figure.
(d) Determining the x- and y-intercepts
The final piece of the puzzle in fully understanding and sketching the graph of the circle is determining the x- and y-intercepts. These intercepts are the points where the circle intersects the x-axis and the y-axis, respectively. To find the x-intercepts, we set y = 0 in the standard form equation of the circle, which is (x - 1/5)^2 + (y - 1/4)^2 = 9/16. Substituting y = 0, we get (x - 1/5)^2 + (0 - 1/4)^2 = 9/16. This simplifies to (x - 1/5)^2 + 1/16 = 9/16. Subtracting 1/16 from both sides gives (x - 1/5)^2 = 8/16, which simplifies to (x - 1/5)^2 = 1/2. Taking the square root of both sides yields x - 1/5 = ±√(1/2). Therefore, x = 1/5 ± √(1/2). This gives us two x-intercepts: x = 1/5 + √(1/2) and x = 1/5 - √(1/2). Approximating these values, we get x ≈ 0.907 and x ≈ -0.507. To find the y-intercepts, we set x = 0 in the standard form equation. Substituting x = 0, we get (0 - 1/5)^2 + (y - 1/4)^2 = 9/16. This simplifies to 1/25 + (y - 1/4)^2 = 9/16. Subtracting 1/25 from both sides gives (y - 1/4)^2 = 9/16 - 1/25. Finding a common denominator, we get (y - 1/4)^2 = 225/400 - 16/400, which simplifies to (y - 1/4)^2 = 209/400. Taking the square root of both sides yields y - 1/4 = ±√(209/400). Therefore, y = 1/4 ± √(209/400). This gives us two y-intercepts: y = 1/4 + √(209/400) and y = 1/4 - √(209/400). Approximating these values, we get y ≈ 0.612 and y ≈ -0.112. The x- and y-intercepts provide additional key points that help us accurately sketch the circle's graph. They show where the circle crosses the axes, giving us a better understanding of its position in the coordinate plane. These intercepts, along with the center and radius, provide a complete picture of the circle's properties. Determining the intercepts is a valuable step in the process of graphing circles, as it enhances the accuracy and clarity of the visual representation.
By following these steps – rewriting the equation, completing the square, determining the center and radius, graphing the circle, and finding the intercepts – we can confidently tackle any circle equation presented in general form. This process not only allows us to visualize the circle but also deepens our understanding of the relationship between algebraic equations and geometric shapes.