Circle Equation Identifying Equation For Center (-5 5) Radius 3

At the heart of analytical geometry lies the equation of a circle, a fundamental concept that allows us to precisely define and represent circles in the coordinate plane. This equation serves as a powerful tool for various applications in mathematics, physics, engineering, and computer graphics. To truly grasp the intricacies of circles, it's crucial to understand the standard form equation and how it relates to the circle's center and radius. The standard form equation of a circle is given by: (x - h)^2 + (y - k)^2 = r^2 where (h, k) represents the coordinates of the center of the circle, and r denotes the radius. This equation stems from the Pythagorean theorem, which relates the distances between points in a right-angled triangle. By applying this theorem to the circle, we can express the relationship between the coordinates of any point on the circle (x, y), the center (h, k), and the radius r. The equation essentially states that the square of the distance between any point on the circle and the center is always equal to the square of the radius. This distance is calculated using the distance formula, which is derived from the Pythagorean theorem. To derive the equation, consider a circle with center (h, k) and radius r. Let (x, y) be any point on the circumference of the circle. The distance between (x, y) and (h, k) can be calculated as √((x - h)^2 + (y - k)^2). By definition, this distance is equal to the radius r. Squaring both sides of the equation √((x - h)^2 + (y - k)^2) = r yields the standard form equation (x - h)^2 + (y - k)^2 = r^2. This equation holds true for all points (x, y) on the circle, and it forms the basis for many geometrical and analytical problems involving circles. It is essential to understand how changes in the center coordinates (h, k) and the radius r affect the circle's position and size on the coordinate plane. For example, increasing the radius will enlarge the circle while keeping the center fixed, and shifting the center coordinates will move the circle without changing its size. The standard form equation provides a clear and concise way to capture these geometric properties mathematically, making it an indispensable tool in various fields of study and application.

The question at hand presents a scenario where we need to identify the equation that accurately represents a circle with a specific center and radius. The given information is that the circle has its center at the point (-5, 5) and a radius of 3 units. To solve this problem, we must carefully apply the standard form equation of a circle and substitute the given values. The standard form equation, as previously discussed, is (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center and r is the radius. Our goal is to find the equation that matches the given center and radius. We know that the center is (-5, 5), which means h = -5 and k = 5. The radius is given as 3 units, so r = 3. Now we substitute these values into the standard form equation. Substituting h = -5 into the (x - h)^2 term, we get (x - (-5))^2, which simplifies to (x + 5)^2. Similarly, substituting k = 5 into the (y - k)^2 term, we get (y - 5)^2. Finally, substituting r = 3 into the r^2 term, we get 3^2, which equals 9. Combining these substitutions, we get the equation (x + 5)^2 + (y - 5)^2 = 9. This equation represents a circle with the specified center (-5, 5) and radius 3. To confirm that this is the correct equation, we can analyze the other options provided in the question and identify why they are incorrect. For instance, an equation with (x - 5)^2 would indicate a center with an x-coordinate of 5, which is the opposite sign of what is given. Similarly, an equation with (y + 5)^2 would indicate a center with a y-coordinate of -5. Furthermore, the value on the right side of the equation represents the square of the radius. An equation with 3 on the right side would imply a radius of √3, not 3. By carefully examining each component of the equation, we can ensure that the equation matches the given center and radius and confidently select the correct answer. This process of substitution and verification is crucial in solving problems related to circles and other geometric figures.

Let's now meticulously examine each of the provided options to pinpoint the equation that accurately represents a circle with a center at (-5, 5) and a radius of 3 units. Our guiding principle will be the standard form equation of a circle: (x - h)^2 + (y - k)^2 = r^2. Remember, (h, k) represents the center, and r is the radius. Option A: (x + 5)^2 + (y - 5)^2 = 3. This equation seems promising as it includes (x + 5)^2 and (y - 5)^2, which correspond to the correct center coordinates of (-5, 5). However, the right side of the equation is 3, which means r^2 = 3. This implies that the radius r would be √3, not 3 as specified in the problem. Therefore, Option A is incorrect. Option B: (x + 5)^2 + (y - 5)^2 = 9. This equation also has the correct terms for the center: (x + 5)^2 and (y - 5)^2. The right side of the equation is 9, which means r^2 = 9. Taking the square root of both sides, we find that r = 3. This equation perfectly matches the given center (-5, 5) and radius 3. So, Option B appears to be the correct answer. Option C: (x + 5)^2 + (y - 5)^2 = 6. The left side of this equation correctly represents the center (-5, 5). However, the right side is 6, which implies that r^2 = 6. This gives a radius of √6, which is not equal to the given radius of 3. Therefore, Option C is incorrect. Option D: (x - 5)^2 + (y + 5)^2 = 9. In this equation, the terms (x - 5)^2 and (y + 5)^2 indicate a center at (5, -5), which is not the same as the given center (-5, 5). Although the right side is 9, giving a radius of 3, the incorrect center invalidates this option. Thus, Option D is incorrect. Option E: (x - 5)^2 + (y + 5)^2 = 3. This equation suffers from the same issue as Option D. The terms (x - 5)^2 and (y + 5)^2 represent a center at (5, -5), which does not match the given center. Additionally, the right side is 3, implying a radius of √3, not 3. Consequently, Option E is also incorrect. After careful analysis of all the options, it is evident that Option B, (x + 5)^2 + (y - 5)^2 = 9, is the only equation that correctly represents a circle with a center at (-5, 5) and a radius of 3 units. The correct center coordinates and the accurate radius derived from the right side of the equation confirm this conclusion. By systematically evaluating each option, we have identified the correct answer and reinforced our understanding of the circle equation.

After a comprehensive analysis of all the options, the correct answer to the question is Option B: (x + 5)^2 + (y - 5)^2 = 9. This equation precisely represents a circle with its center located at (-5, 5) and a radius of 3 units. Our journey to this conclusion involved a deep dive into the standard form equation of a circle, which is a fundamental concept in analytical geometry. We revisited the equation (x - h)^2 + (y - k)^2 = r^2, where (h, k) denotes the coordinates of the center and r signifies the radius. Applying this knowledge, we systematically examined each option provided in the question. Option B stood out as the only equation that perfectly aligned with the given center and radius. The terms (x + 5)^2 and (y - 5)^2 accurately reflect the center coordinates (-5, 5). The right side of the equation, 9, corresponds to the square of the radius (3^2), confirming that the radius is indeed 3 units. The other options were ruled out for various reasons. Some had incorrect center coordinates, while others had a radius that did not match the specified value. For instance, options with terms like (x - 5)^2 or (y + 5)^2 would indicate a center different from (-5, 5). Equations with a value other than 9 on the right side would imply a radius different from 3. The process of elimination and careful verification of each option underscored the importance of understanding the standard form equation and its components. By correctly substituting the given center and radius into the equation, we were able to identify the correct answer with confidence. This exercise not only reinforces our understanding of circles but also demonstrates the power of analytical geometry in solving geometric problems. The ability to translate geometric properties into algebraic equations and vice versa is a crucial skill in mathematics and its applications in various fields.

The process of identifying the correct equation for a circle with a given center and radius has highlighted several key takeaways. First and foremost, a solid understanding of the standard form equation of a circle, (x - h)^2 + (y - k)^2 = r^2, is essential. This equation serves as the foundation for solving a wide range of problems related to circles. The ability to correctly identify the center coordinates (h, k) and the radius r from the equation is crucial. Remember that the signs in the equation are opposite to the signs of the center coordinates. For example, (x + 5)^2 indicates an x-coordinate of -5, and (y - 5)^2 indicates a y-coordinate of 5. The value on the right side of the equation represents the square of the radius, so taking the square root of this value will give you the radius. Another important takeaway is the value of systematic analysis and elimination. When faced with multiple options, carefully examining each one and eliminating those that do not meet the criteria can lead you to the correct answer. This approach is particularly useful in multiple-choice questions where there may be distractors designed to mislead you. Furthermore, this exercise emphasizes the connection between algebra and geometry. The standard form equation allows us to represent geometric shapes algebraically, and this representation can be used to solve geometric problems using algebraic techniques. This connection between algebra and geometry is a fundamental concept in mathematics and has numerous applications in other fields. For further exploration, you can investigate other properties of circles, such as the equation of a tangent line to a circle, the intersection of circles, and the relationship between circles and other geometric shapes. You can also explore the general form of the equation of a circle and how to convert it to the standard form. Understanding these concepts will deepen your knowledge of circles and their applications in various areas of mathematics and beyond. Practicing with different problems and scenarios will also solidify your understanding and improve your problem-solving skills. The world of circles is vast and fascinating, and there is always more to discover.