Finding The Radius Of A Circle Given Its Equation X² + Y² - 10x + 6y + 18 = 0
Determining the radius of a circle given its equation is a fundamental concept in analytic geometry. In this comprehensive guide, we will explore the equation x² + y² - 10x + 6y + 18 = 0 and systematically unravel the steps to find its radius. We'll not only provide the solution but also delve into the underlying principles and techniques, ensuring a thorough understanding of the topic. This article is crafted to serve as an invaluable resource for students, educators, and anyone with a keen interest in mathematics.
Identifying the Circle Equation
The given equation, x² + y² - 10x + 6y + 18 = 0, is a second-degree equation in two variables, x and y. Recognizing this as a circle equation is the first step. The general form of a circle's equation is (x - h)² + (y - k)² = r², where (h, k) represents the center of the circle and r denotes the radius. Our mission is to transform the given equation into this standard form. This transformation will involve a process called completing the square, a powerful algebraic technique that allows us to rewrite quadratic expressions in a more manageable form. Understanding the general form is crucial because it directly reveals the circle's center and radius, which are key parameters for characterizing the circle.
The ability to identify a circle equation from its general form is a crucial skill in coordinate geometry. The presence of both x² and y² terms with equal coefficients (in this case, both are 1) is a strong indicator of a circle. However, we must also ensure that the equation can be manipulated into the standard form without leading to contradictions or imaginary solutions. This involves a careful examination of the coefficients and the constant term, as well as the application of algebraic techniques to reveal the underlying geometric properties. By mastering this identification process, we lay the foundation for solving a wide range of problems involving circles and their equations.
Furthermore, recognizing the circle equation's structure allows us to visualize the geometric object it represents. A circle is defined as the set of all points equidistant from a central point, and this distance is the radius. The equation captures this geometric definition in an algebraic form, providing a powerful tool for analysis and manipulation. By understanding the relationship between the equation and the circle's properties, we can solve problems such as finding the center, radius, tangent lines, and intersections with other geometric figures. This connection between algebra and geometry is a fundamental theme in mathematics, and mastering it is essential for success in higher-level studies.
Completing the Square: A Step-by-Step Approach
To convert the given equation x² + y² - 10x + 6y + 18 = 0 into the standard form, we employ the method of completing the square. This involves rearranging the terms and adding specific constants to both sides of the equation to create perfect square trinomials. Let's break down the process step-by-step:
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Group the x and y terms:
Begin by grouping the x terms and the y terms together: (x² - 10x) + (y² + 6y) + 18 = 0. This rearrangement sets the stage for completing the square for each variable separately.
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Move the constant term to the right side:
Subtract 18 from both sides of the equation to isolate the variable terms: (x² - 10x) + (y² + 6y) = -18. This step prepares the equation for the addition of constants that will complete the squares.
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Complete the square for x:
To complete the square for x² - 10x, take half of the coefficient of the x term (-10), square it ((-5)² = 25), and add it to both sides of the equation. This gives us: (x² - 10x + 25) + (y² + 6y) = -18 + 25. The expression x² - 10x + 25 is now a perfect square trinomial, which can be factored as (x - 5)².
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Complete the square for y:
Similarly, to complete the square for y² + 6y, take half of the coefficient of the y term (6), square it ((3)² = 9), and add it to both sides of the equation. This leads to: (x² - 10x + 25) + (y² + 6y + 9) = -18 + 25 + 9. The expression y² + 6y + 9 is now a perfect square trinomial, which can be factored as (y + 3)².
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Rewrite the equation in standard form:
Rewrite the perfect square trinomials as squared binomials and simplify the right side of the equation: (x - 5)² + (y + 3)² = 16. This is now the standard form of the circle equation, (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius.
Completing the square is a versatile technique applicable to various quadratic equations, not just those representing circles. It allows us to transform quadratic expressions into a form that reveals key information, such as the vertex of a parabola or the center and radius of a circle. The underlying principle is to manipulate the equation algebraically while maintaining equality, ultimately leading to a more insightful representation of the underlying geometric or algebraic structure. Mastering this technique is essential for solving a wide range of mathematical problems.
Identifying the Radius
Now that we have the equation in standard form, (x - 5)² + (y + 3)² = 16, identifying the radius is straightforward. Recall that the standard form of a circle's equation is (x - h)² + (y - k)² = r², where r is the radius. By comparing our equation to the standard form, we can see that r² = 16. To find the radius r, we simply take the square root of both sides of the equation.
Taking the square root of 16 gives us r = √16 = 4. Since the radius is a measure of distance, we consider only the positive square root. Therefore, the radius of the circle is 4 units. This result corresponds to option B in the given choices. The radius, in essence, defines the size of the circle, representing the constant distance from any point on the circle to its center. Understanding how to extract the radius from the standard equation is crucial for solving various problems related to circles, such as calculating circumference, area, and determining the circle's position in the coordinate plane.
The radius is a fundamental parameter of a circle, and its value dictates many other properties of the circle. For instance, the diameter, which is twice the radius, determines the overall span of the circle. The circumference, calculated as 2πr, measures the distance around the circle, and the area, given by πr², quantifies the space enclosed by the circle. Therefore, knowing the radius allows us to compute these other important characteristics of the circle. Furthermore, the radius plays a critical role in determining the circle's relationship with other geometric figures, such as lines, triangles, and other circles. By analyzing the radius, we can determine whether two circles intersect, whether a line is tangent to a circle, or whether a circle is inscribed within a polygon. This makes the radius a central concept in geometric analysis and problem-solving.
Conclusion: The Radius Unveiled
In conclusion, by transforming the given equation x² + y² - 10x + 6y + 18 = 0 into the standard form (x - 5)² + (y + 3)² = 16 through the method of completing the square, we successfully determined that the radius of the circle is 4 units. This process not only provides the solution but also reinforces our understanding of circle equations and the powerful technique of completing the square. This comprehensive approach ensures that we can confidently tackle similar problems in the future.
The ability to manipulate equations and extract geometric information is a hallmark of mathematical proficiency. The process we've followed—from recognizing the circle equation to completing the square and identifying the radius—demonstrates the interconnectedness of algebra and geometry. This interplay between algebraic manipulation and geometric interpretation is a recurring theme in mathematics, and mastering it opens doors to more advanced concepts and problem-solving techniques. By understanding the underlying principles and practicing these techniques, we can develop a deeper appreciation for the elegance and power of mathematics.
Moreover, the skills acquired in this exercise extend beyond the specific problem of finding the radius of a circle. Completing the square, for example, is a versatile technique that finds applications in various areas of mathematics, including solving quadratic equations, graphing parabolas, and working with conic sections. The ability to transform equations into more informative forms is a valuable asset in any mathematical endeavor. Therefore, the time and effort invested in mastering these fundamental concepts and techniques will yield significant returns in future studies and problem-solving endeavors.
By understanding the radius and its significance, we gain a deeper appreciation for the geometry of circles and their role in mathematics and the world around us. Circles are ubiquitous in nature and engineering, from the orbits of planets to the design of wheels and gears. Their simple yet elegant properties make them a fundamental shape in geometry, and understanding their equations and characteristics is essential for a wide range of applications. Therefore, the knowledge and skills we've gained in this exploration will serve us well in future mathematical pursuits and real-world problem-solving.
