Finding The Radius Of A Circle Given Its Equation $x^2+y^2-2x+8y-47=0$

Jul 13, 2025 by Admin 71 views

$Unveiling the Radius of a Circle The radius of a circle is a fundamental property that dictates its size and is crucial in various geometric calculations and applications. When a circle's equation is given in the general form, extracting the radius requires a systematic approach involving completing the square. This article delves into the process of determining the radius of a circle from its equation, specifically focusing on the equation ${x^2 + y^2 - 2x + 8y - 47 = 0}$. We will explore the underlying principles, step-by-step methods, and practical implications of this essential geometric concept. Understanding the radius of a circle not only enhances our comprehension of circular geometry but also equips us with the tools to solve a wide range of problems in mathematics, physics, and engineering. ## Understanding the General Equation of a Circle To effectively determine the radius, we must first understand the general equation of a circle and how it relates to the circle's center and radius. The general equation of a circle is given by: ${x^2 + y^2 + 2gx + 2fy + c = 0}$ where ${(-g, -f)}$ represents the center of the circle, and ${r}$ represents the radius. This equation is derived from the standard form of a circle's equation, which is: ${(x - h)^2 + (y - k)^2 = r^2}$ where ${(h, k)}$ is the center and ${r}$ is the radius. The general form is obtained by expanding the standard form and rearranging the terms. To find the radius from the general equation, we need to convert it back into the standard form. This is achieved by a technique called completing the square. Completing the square allows us to rewrite quadratic expressions in a form that reveals the center and radius of the circle. The process involves manipulating the equation to create perfect square trinomials for both the ${x}$ and ${y}$ terms. By understanding the relationship between the general and standard forms, we can confidently extract the radius of any circle given its equation. This knowledge forms the foundation for solving a variety of geometric problems and understanding circular phenomena in various fields. ## Step-by-Step Solution: Finding the Radius Let's apply the method of completing the square to the given equation ${x^2 + y^2 - 2x + 8y - 47 = 0}$ to find the radius. 1. **Group the x and y terms:** Begin by rearranging the equation to group the ${x}$ terms and the ${y}$ terms together: ${(x^2 - 2x) + (y^2 + 8y) = 47}$ This rearrangement sets the stage for completing the square for both ${x}$ and ${y}$ variables. 2. **Complete the square for x:** To complete the square for the ${x}$ terms, we need to add and subtract the square of half the coefficient of the ${x}$ term. The coefficient of ${x}$ is -2, so half of it is -1, and its square is 1. Add and subtract 1 within the ${x}$ terms: ${(x^2 - 2x + 1) - 1}$ The expression ${x^2 - 2x + 1}$ is now a perfect square trinomial, which can be factored as ${(x - 1)^2}$. 3. **Complete the square for y:** Similarly, for the ${y}$ terms, we need to add and subtract the square of half the coefficient of the ${y}$ term. The coefficient of ${y}$ is 8, so half of it is 4, and its square is 16. Add and subtract 16 within the ${y}$ terms: ${(y^2 + 8y + 16) - 16}$ The expression ${y^2 + 8y + 16}$ is a perfect square trinomial, which can be factored as ${(y + 4)^2}$. 4. **Rewrite the equation:** Substitute the completed square expressions back into the equation: ${(x - 1)^2 - 1 + (y + 4)^2 - 16 = 47}$ 5. **Simplify the equation:** Combine the constant terms and move them to the right side of the equation: ${(x - 1)^2 + (y + 4)^2 = 47 + 1 + 16}$ ${(x - 1)^2 + (y + 4)^2 = 64}$ 6. **Identify the radius:** Now the equation is in the standard form ${(x - h)^2 + (y - k)^2 = r^2}$, where ${r^2 = 64}$. Taking the square root of both sides, we get: $\r = \sqrt{64} = 8\)$ Therefore, the radius of the circle is 8 units. ## Practical Implications of the Radius The radius of a circle is not just a number; it holds significant practical implications across various fields. In **geometry**, the radius is essential for calculating the circumference and area of a circle. The circumference, given by ${2\pi r}$, determines the distance around the circle, while the area, given by ${\pi r^2}$, determines the space enclosed by the circle. These calculations are fundamental in many geometric problems and constructions. In **engineering**, the radius is crucial in designing circular structures, such as wheels, pipes, and gears. The dimensions of these components directly impact their performance and functionality. For instance, the radius of a gear determines its speed and torque transmission capabilities. Similarly, the radius of a pipe affects its flow capacity. In **physics**, the radius plays a vital role in understanding circular motion and rotational dynamics. The radius is a key parameter in calculating the centripetal force, angular velocity, and moment of inertia of rotating objects. These concepts are essential in fields like mechanics, astrophysics, and electromagnetism. In **everyday life**, the radius is relevant in numerous applications. From the size of pizzas and cakes to the design of circular gardens and swimming pools, the radius helps us determine dimensions and quantities. Understanding the radius allows us to make informed decisions and solve practical problems in our daily routines. ## Common Mistakes and How to Avoid Them When working with circle equations and radii, several common mistakes can occur. Recognizing these pitfalls and understanding how to avoid them is crucial for accurate problem-solving. 1. **Incorrectly completing the square:** A frequent error is making mistakes while completing the square. This often involves miscalculating the term to be added and subtracted or incorrectly factoring the perfect square trinomial. **How to avoid it:** Double-check each step of the completing the square process. Ensure that you are adding and subtracting the correct value (half the coefficient of the linear term, squared) and that you are factoring the trinomial correctly. Practice with various examples to build confidence and accuracy. 2. **Forgetting to take the square root:** After completing the square and obtaining the value of ${r^2}$, some individuals forget to take the square root to find the radius ${r}$. **How to avoid it:** Always remember that the equation represents ${r^2}$, not ${r}$. Make it a habit to take the square root as the final step in determining the radius. 3. **Misinterpreting the equation:** Another mistake is misinterpreting the signs in the standard form of the circle equation. For example, confusing ${(x - h)^2}$ with ${(x + h)^2}$ can lead to incorrect identification of the center and, consequently, the radius. **How to avoid it:** Pay close attention to the signs in the equation. Remember that the standard form is ${(x - h)^2 + (y - k)^2 = r^2}$, so a ${+}$ sign inside the parentheses indicates a negative coordinate for the center, and vice versa. 4. **Algebraic errors:** Simple algebraic errors, such as incorrect addition, subtraction, or distribution, can lead to incorrect results. **How to avoid it:** Work carefully and methodically, writing down each step clearly. Double-check your calculations and consider using a calculator for complex arithmetic. By being mindful of these common mistakes and implementing strategies to avoid them, you can significantly improve your accuracy and confidence in solving circle-related problems. ## Conclusion In conclusion, finding the radius of a circle from its general equation involves a systematic approach using the method of completing the square. By rearranging the equation, grouping the ${x}$ and ${y}$ terms, and completing the square for each variable, we can transform the equation into the standard form ${(x - h)^2 + (y - k)^2 = r^2}$. This form directly reveals the center ${(h, k)}$ and the radius ${r}$ of the circle. In the specific case of the equation ${x^2 + y^2 - 2x + 8y - 47 = 0}$, we demonstrated the step-by-step process of completing the square, which led us to the standard form ${(x - 1)^2 + (y + 4)^2 = 64}$. Taking the square root of 64, we found that the radius of the circle is 8 units. Understanding the radius of a circle is fundamental in geometry and has practical applications in various fields, including engineering, physics, and everyday life. By mastering the technique of completing the square and avoiding common mistakes, we can confidently determine the radius of any circle given its equation and apply this knowledge to solve a wide range of problems. This skill not only enhances our mathematical proficiency but also deepens our appreciation for the elegance and utility of circular geometry.$