Finding The Equation Of A Circle A Comprehensive Guide
At the heart of coordinate geometry lies the equation of a circle, a fundamental concept that describes the set of all points equidistant from a central point. The standard equation of a circle 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 stems directly from the Pythagorean theorem, which relates the sides of a right triangle. In the context of a circle, the radius serves as the hypotenuse, while the horizontal and vertical distances from any point on the circle to the center form the other two sides. Understanding this equation is the cornerstone for solving problems involving circles, including determining their equations given specific information.
In this particular problem, we are given the coordinates of the center of circle C as (-2, 10) and a point P(10, 5) that lies on the circle. Our mission is to determine the equation that accurately represents circle C. To achieve this, we need to find the radius of the circle, which is the distance between the center and any point on the circle. Once we have the radius, we can plug the center coordinates and the radius into the standard equation of a circle to obtain the desired equation. This process involves applying the distance formula, a direct application of the Pythagorean theorem, to calculate the distance between two points in a coordinate plane. By carefully substituting the given values and performing the necessary calculations, we can arrive at the correct equation that describes circle C.
The distance formula, derived from the Pythagorean theorem, is our tool for finding the radius of circle C. Given two points (x1, y1) and (x2, y2), the distance between them is calculated as:
√[(x2 - x1)^2 + (y2 - y1)^2]
In our case, the center of circle C is (-2, 10) and point P on the circle is (10, 5). Let's plug these coordinates into the distance formula:
Distance = √[(10 - (-2))^2 + (5 - 10)^2]
Simplifying the expression inside the square root, we get:
Distance = √[(10 + 2)^2 + (5 - 10)^2]
Distance = √[(12)^2 + (-5)^2]
Distance = √(144 + 25)
Distance = √169
Distance = 13
Therefore, the radius of circle C is 13 units. This crucial value will be used in the next step to construct the equation of the circle. Understanding the distance formula and its application is essential not only for finding the radius but also for various other geometric problems involving distances between points. The ability to accurately calculate distances in a coordinate plane is a fundamental skill in coordinate geometry, enabling us to solve a wide range of problems related to geometric shapes and their properties. In this case, the calculated radius will serve as the foundation for building the equation of circle C, allowing us to describe its position and size in the coordinate plane.
Now that we know the center of circle C is (-2, 10) and the radius is 13, we can construct the equation of the circle using the standard form:
(x - h)^2 + (y - k)^2 = r^2
where (h, k) is the center and r is the radius. Substituting the values we found, we get:
(x - (-2))^2 + (y - 10)^2 = 13^2
Simplifying this equation, we have:
(x + 2)^2 + (y - 10)^2 = 169
This is the equation that represents circle C. By plugging in any point (x, y) that lies on the circle into this equation, the equation will hold true. This equation encapsulates the geometric properties of the circle, including its center and radius, allowing us to describe its position and size in the coordinate plane. The process of constructing the equation of a circle involves understanding the relationship between the center, radius, and the coordinates of points on the circle. This understanding is crucial for solving problems involving circles and their properties, including finding the equation of a circle given specific information, determining if a point lies on a circle, and finding the intersection of circles with other geometric shapes.
Comparing our derived equation, (x + 2)^2 + (y - 10)^2 = 169, with the given options, we can see that it matches option D.
Therefore, the correct answer is:
D. (x + 2)^2 + (y - 10)^2 = 169
This confirms that our calculations and understanding of the circle equation are accurate. By carefully applying the distance formula and substituting the values into the standard equation of a circle, we were able to successfully identify the correct equation representing circle C. This process demonstrates the importance of understanding the fundamental concepts of coordinate geometry and their applications in solving geometric problems. The ability to manipulate equations and relate them to geometric properties is a key skill in mathematics, allowing us to analyze and describe shapes and their relationships in a coordinate plane.
In conclusion, we successfully determined the equation of circle C by applying the distance formula to find the radius and then substituting the center coordinates and radius into the standard equation of a circle. This problem highlights the importance of understanding the relationship between the center, radius, and equation of a circle. By mastering these concepts, you can confidently solve a wide range of problems involving circles in coordinate geometry. The ability to manipulate equations and connect them to geometric properties is a fundamental skill in mathematics, paving the way for tackling more complex problems in geometry and related fields. The key takeaways from this exercise include the importance of the distance formula, the standard equation of a circle, and the ability to substitute values accurately to arrive at the correct solution. With a solid understanding of these concepts, you can confidently approach and solve problems involving circles and their equations.
Understanding circle equations is crucial in various fields, from geometry and calculus to computer graphics and physics. The ability to determine the equation of a circle given certain parameters allows us to model and analyze circular shapes and their properties. This skill is essential for solving real-world problems, such as determining the trajectory of a projectile, designing circular structures, and understanding the motion of objects in circular paths. By mastering the concepts and techniques presented in this article, you will be well-equipped to tackle a wide range of problems involving circles and their applications.
