Equation Of A Circle Finding The Equation Of Circle C
In this article, we will delve into the process of finding the equation of a circle given its center and a point it passes through. Specifically, we'll tackle the problem of circle C, which has its center at the coordinates (-2, 10) and includes the point P(10, 5). This is a fundamental concept in analytic geometry, crucial for understanding the relationship between geometric shapes and algebraic equations. Understanding how to derive the equation of a circle is essential for various applications in mathematics, physics, and engineering. This article provides a step-by-step guide, ensuring clarity and comprehension for students and enthusiasts alike. By the end of this discussion, you will have a firm grasp on how to approach such problems and confidently determine the equation of any circle given its center and a point on its circumference.
Understanding the Standard Equation of a Circle
Before we dive into the specifics of circle C, let's first establish a solid understanding of the standard equation of a circle. This equation is the foundation upon which we'll build our solution. The standard equation of a circle with center (h, k) and radius r is given by:
(x - h)² + (y - k)² = r²
Here, (x, y) represents any point on the circumference of the circle. The values h and k define the center's coordinates, and r represents the radius, which is the distance from the center to any point on the circle. This equation stems from the Pythagorean theorem, which relates the sides of a right triangle. Imagine a right triangle formed by the horizontal distance (x - h), the vertical distance (y - k), and the radius r as the hypotenuse. The Pythagorean theorem then directly translates into the circle's equation. Grasping this foundational principle is vital, as it allows us to translate geometric properties into algebraic expressions, which is a core concept in coordinate geometry. By understanding this equation, we can easily represent circles algebraically and solve problems involving their properties. Furthermore, this equation forms the basis for understanding other conic sections, such as ellipses and hyperbolas, making it a cornerstone of higher-level mathematics.
Calculating the Radius of Circle C
The problem states that circle C has a center at (-2, 10) and passes through the point P(10, 5). To determine the equation of the circle, we first need to find its radius. The radius, as we know, is the distance between the center of the circle and any point on its circumference. In this case, we can calculate the distance between the center (-2, 10) and the point P(10, 5) using the distance formula, which is derived from the Pythagorean theorem. The distance formula is given by:
r = √[(x₂ - x₁)² + (y₂ - y₁)²]
Where (x₁, y₁) and (x₂, y₂) are the coordinates of the two points. Plugging in the coordinates of the center (-2, 10) and the point P(10, 5) into the distance formula, we get:
r = √[(10 - (-2))² + (5 - 10)²] r = √[(12)² + (-5)²] r = √(144 + 25) r = √169 r = 13
Therefore, the radius of circle C is 13 units. This calculation is a critical step because the radius is a key component of the circle's equation. Without knowing the radius, we cannot fully define the circle algebraically. The distance formula is a fundamental tool in coordinate geometry and is used extensively in various mathematical and scientific applications. Mastering this formula is essential for solving problems involving distances between points in a coordinate plane. Now that we have determined the radius, we are one step closer to finding the equation of circle C. The next step involves substituting the center coordinates and the radius into the standard equation of a circle.
Deriving the Equation of Circle C
Now that we have calculated the radius of circle C to be 13 units and we know the center is at (-2, 10), we can plug these values into the standard equation of a circle, which is:
(x - h)² + (y - k)² = r²
Where (h, k) is the center and r is the radius. Substituting h = -2, k = 10, and r = 13 into the equation, we get:
(x - (-2))² + (y - 10)² = 13²
Simplifying this equation, we have:
(x + 2)² + (y - 10)² = 169
This is the equation of circle C. It represents all the points (x, y) that lie on the circumference of the circle with the given center and radius. The equation encapsulates the geometric properties of the circle in an algebraic form, allowing us to analyze and manipulate it using algebraic techniques. This process of deriving the equation highlights the powerful connection between geometry and algebra. By understanding the standard equation of a circle and knowing how to calculate the radius, we can easily determine the equation of any circle given its center and a point on its circumference. This skill is invaluable in various mathematical contexts, including graphing circles, solving geometric problems, and understanding more complex geometric shapes. The derived equation, (x + 2)² + (y - 10)² = 169, completely defines circle C and allows us to further explore its properties.
Analyzing the Answer Choices
Now that we've derived the equation of circle C as (x + 2)² + (y - 10)² = 169, let's compare it with the given answer choices to identify the correct option. The answer choices typically present variations in the signs of the center coordinates and the value of the radius squared. This step is crucial to ensure that we have correctly applied the standard equation and made no errors in our calculations. By carefully comparing our derived equation with the options, we can reinforce our understanding of the equation's structure and the significance of each term. The correct answer choice will have the same terms and values as our derived equation, including the correct signs for the center coordinates and the correct value for the radius squared. This comparative analysis not only helps us to select the right answer but also strengthens our ability to recognize and interpret the equation of a circle. Furthermore, it provides an opportunity to identify common mistakes, such as incorrect sign usage or miscalculation of the radius, which can be avoided in future problems. The process of analyzing answer choices is a vital skill in problem-solving and helps to develop a more thorough understanding of the underlying concepts.
Conclusion: The Equation of Circle C
In conclusion, after carefully analyzing the problem and applying the standard equation of a circle, we have successfully determined the equation of circle C. By calculating the radius using the distance formula and substituting the center coordinates and radius into the standard equation, we arrived at the equation:
(x + 2)² + (y - 10)² = 169
This equation accurately represents the circle with its center at (-2, 10) and passing through the point (10, 5). This exercise demonstrates the importance of understanding fundamental concepts in coordinate geometry, such as the standard equation of a circle and the distance formula. Mastering these concepts allows us to solve a wide range of problems involving circles and other geometric shapes. Furthermore, this process highlights the interconnectedness of geometry and algebra, showcasing how algebraic equations can be used to represent and analyze geometric figures. The ability to derive the equation of a circle given its center and a point on its circumference is a valuable skill in mathematics and has applications in various fields, including physics, engineering, and computer graphics. This article has provided a comprehensive guide to solving this type of problem, ensuring that readers can confidently approach similar challenges in the future. The key takeaway is the understanding of the standard equation, the ability to calculate the radius, and the careful substitution of values to obtain the correct equation.
