Equation Of A Circle With Center (2,-6) Passing Through (6,-1)

Introduction to Circle Equations

The fundamental concept required to find an equation of a circle involves understanding its standard form. The equation of a circle in the Cartesian plane is expressed as (x - h)^2 + (y - k)^2 = r^2, where (h, k) represents the coordinates of the circle's center, and r denotes its radius. This equation arises directly from the Pythagorean theorem and the definition of a circle as the set of all points equidistant from a central point. To effectively utilize this formula, one needs to identify the center coordinates and the radius. The center provides the (h, k) values directly, while the radius can be determined if another point on the circle is known, as it represents the distance between this point and the center. Understanding this equation is crucial not just for academic problem-solving but also for various real-world applications, including navigation systems, computer graphics, and engineering designs. For instance, in navigation, knowing the equation of a circular path can help in calculating distances and bearings. In computer graphics, circles are fundamental elements in creating images and animations. In engineering, circular shapes are used in designing various mechanical components and structures. Therefore, a solid grasp of the circle equation is invaluable in numerous fields, making it a worthwhile subject of study and practice. The equation's simplicity belies its power, as it succinctly captures the essence of a circle's geometry. By varying the parameters h, k, and r, one can describe any circle in the plane, highlighting the equation's versatility and fundamental nature in mathematics and its applications.

Determining the Center and Radius

In this specific problem, we're tasked with finding the equation of a circle given its center at (2, -6) and a point it passes through, (6, -1). The center coordinates provide us with the (h, k) values directly, where h = 2 and k = -6. These values are essential as they form the foundation of the circle's equation. The remaining unknown is the radius, r, which represents the distance from the center to any point on the circle. Since we know the center and a point on the circle, we can use the distance formula to calculate the radius. The distance formula, derived from the Pythagorean theorem, states that the distance between two points (x1, y1) and (x2, y2) in the plane is given by √((x2 - x1)^2 + (y2 - y1)^2). Applying this formula to our problem, we find the distance between the center (2, -6) and the point (6, -1). This distance will be the radius of our circle. Substituting the coordinates into the formula, we get r = √((6 - 2)^2 + (-1 - (-6))^2). Simplifying this expression will give us the numerical value of the radius. It's important to note that the radius is a crucial parameter that dictates the size of the circle. A larger radius implies a larger circle, while a smaller radius indicates a smaller circle. The radius, along with the center coordinates, uniquely defines a circle in the plane. Once we have determined the radius, we will have all the necessary components to write the equation of the circle in its standard form. This step-by-step approach, from identifying the known values to applying the appropriate formula, is a fundamental problem-solving technique in mathematics. By breaking down the problem into smaller, manageable steps, we can systematically arrive at the solution.

Calculating the Radius

To calculate the radius, we apply the distance formula: r = √((6 - 2)^2 + (-1 - (-6))^2). First, we simplify the expressions within the parentheses. Subtracting 2 from 6 gives us 4, so the first term becomes 4^2. For the second term, subtracting -6 from -1 is equivalent to adding 6 to -1, which results in 5. Thus, the second term becomes 5^2. Now, we have r = √(4^2 + 5^2). Next, we compute the squares: 4^2 is 16, and 5^2 is 25. Substituting these values, we get r = √(16 + 25). Adding 16 and 25 gives us 41, so now we have r = √41. The square root of 41 is an irrational number, meaning it cannot be expressed as a simple fraction. Therefore, we leave the radius in its exact form, √41, for the most accurate result. This ensures that our final equation is precise and avoids any rounding errors. Understanding how to work with square roots and irrational numbers is a key skill in algebra and geometry. It allows us to represent quantities exactly, without relying on approximations. In many mathematical and scientific contexts, maintaining precision is crucial, and leaving values in their exact form is the preferred approach. In our case, using √41 as the radius will ensure that the circle equation we derive accurately represents the given conditions. Now that we have the radius and the center coordinates, we are ready to assemble the equation of the circle. This involves substituting the values of h, k, and r into the standard form of the circle equation.

Forming the Equation of the Circle

With the center (h, k) = (2, -6) and the radius r = √41, we can now form the equation of the circle. Recall the standard form of a circle's equation: (x - h)^2 + (y - k)^2 = r^2. We substitute the values of h, k, and r into this equation. Replacing h with 2, k with -6, and r with √41, we get: (x - 2)^2 + (y - (-6))^2 = (√41)^2. Next, we simplify the equation. The term (y - (-6)) can be rewritten as (y + 6). Squaring the square root of 41, (√41)^2, gives us 41. So, the equation becomes: (x - 2)^2 + (y + 6)^2 = 41. This is the equation of the circle in standard form. It clearly shows the center of the circle at (2, -6) and the radius as √41. The standard form of the equation is particularly useful because it directly reveals these key characteristics of the circle. It allows for easy identification of the center and radius, which are essential for graphing the circle and understanding its properties. This equation concisely describes the circle that meets the given conditions: it has a center at (2, -6) and passes through the point (6, -1). The equation we have derived is not only a mathematical expression but also a complete representation of the circle in the Cartesian plane. It encapsulates all the information needed to fully describe the circle's position and size. Understanding how to manipulate and interpret such equations is a fundamental skill in analytic geometry and is widely applicable in various mathematical and scientific contexts.

Final Equation and Conclusion

Therefore, the final equation of the circle with center (2, -6) and passing through (6, -1) is (x - 2)^2 + (y + 6)^2 = 41. This equation represents all the points (x, y) that lie on the circle. To summarize, we started with the given information: the center of the circle and a point it passes through. We then used the distance formula to calculate the radius, which turned out to be √41. Finally, we substituted the center coordinates and the radius into the standard form of the circle equation to arrive at our final answer. This process demonstrates a systematic approach to solving geometry problems, where we identify the knowns, apply relevant formulas, and simplify the resulting expressions. The equation we have obtained provides a complete and concise description of the circle. It can be used to verify whether a given point lies on the circle, to graph the circle, or to further explore its geometric properties. Understanding the relationship between the geometric properties of a circle and its algebraic equation is a fundamental concept in mathematics. It bridges the gap between visual representations and symbolic expressions, allowing us to analyze and manipulate geometric objects using algebraic techniques. This interplay between geometry and algebra is a powerful tool in problem-solving and is essential for further studies in mathematics and related fields. The ability to derive and interpret equations of geometric shapes is a cornerstone of mathematical literacy and has wide-ranging applications in science, engineering, and technology. The equation (x - 2)^2 + (y + 6)^2 = 41 serves as a succinct and precise mathematical description of the circle we set out to define.