Equation Of Circle With Center (7,-24) Passing Through Origin
In this article, we will delve into the process of identifying the equation of a circle given its center and a point it passes through. Specifically, we'll focus on a circle with its center at the point (7, -24) and passing through the origin (0, 0). This problem is a classic example of applying the standard equation of a circle and the distance formula. Understanding these concepts is crucial for various applications in geometry, calculus, and even physics. Let's explore the steps involved in solving this problem and solidify our understanding of circles and their equations.
Understanding the Standard Equation of a Circle
To begin, let's understand the fundamental concept: the standard equation of a circle. The standard equation of a circle with center (h, k) and radius r is given by:
(x - h)² + (y - k)² = r²
This equation is derived from the Pythagorean theorem and represents the set of all points (x, y) that are a distance r away from the center (h, k). Understanding this equation is the cornerstone of solving problems related to circles. Each term in the equation plays a critical role in defining the circle's position and size in the coordinate plane. The (x - h)² and (y - k)² terms represent the squared horizontal and vertical distances from any point on the circle to the center, while r² represents the squared radius of the circle. By manipulating this equation, we can determine various properties of the circle, such as its center, radius, and points that lie on its circumference.
In our problem, we are given the center of the circle as (7, -24). This means that h = 7 and k = -24. We also know that the circle passes through the origin (0, 0). This information is crucial because it allows us to determine the radius of the circle. The radius is the distance between the center and any point on the circle. Since we know the center and a point on the circle (the origin), we can use the distance formula to find the radius. Once we have the radius, we can plug all the values into the standard equation of a circle and obtain the equation for the specific circle in this problem.
Calculating the Radius Using the Distance Formula
To find the equation of the circle, we first need to determine its radius. The radius is the distance between the center of the circle (7, -24) and the origin (0, 0). We can calculate this distance using the distance formula, which is derived from the Pythagorean theorem:
Distance = √[(x₂ - x₁)² + (y₂ - y₁)²]
In our case, (x₁, y₁) = (7, -24) and (x₂, y₂) = (0, 0). Plugging these values into the distance formula, we get:
Distance = √[(0 - 7)² + (0 - (-24))²]
Distance = √[(-7)² + (24)²]
Distance = √[49 + 576]
Distance = √625
Distance = 25
Therefore, the radius of the circle is 25. This radius is a critical parameter in defining the circle's size. A larger radius means a larger circle, and vice versa. The distance formula is a fundamental tool in coordinate geometry, allowing us to calculate the distance between any two points in the coordinate plane. It is used extensively in various mathematical and scientific applications, including finding the lengths of line segments, determining the distance between objects in space, and calculating the magnitude of vectors.
With the radius calculated, we now have all the necessary information to write the equation of the circle. We know the center (h, k) is (7, -24) and the radius r is 25. The next step is to substitute these values into the standard equation of a circle and simplify to obtain the final equation.
Plugging the Values into the Standard Equation
Now that we have determined the radius to be 25, we can substitute the values of the center (h = 7, k = -24) and the radius (r = 25) into the standard equation of a circle:
(x - h)² + (y - k)² = r²
Substituting the values, we get:
(x - 7)² + (y - (-24))² = 25²
(x - 7)² + (y + 24)² = 625
This equation represents the circle with center (7, -24) and radius 25. The equation clearly shows the relationship between the x and y coordinates of any point on the circle and the circle's center and radius. This form of the equation is particularly useful because it directly reveals the center and radius of the circle, making it easy to visualize and analyze the circle's properties. The left-hand side of the equation represents the squared distance from any point (x, y) on the circle to the center (7, -24), while the right-hand side represents the squared radius.
By understanding how to manipulate this equation, we can solve a wide range of problems related to circles, such as finding the intersection points of a circle with a line, determining the equation of a tangent line to a circle, and calculating the area and circumference of a circle. The standard equation of a circle is a fundamental tool in geometry and is essential for anyone studying circles and their properties.
Identifying the Correct Option
By substituting the values into the standard equation, we found the equation of the circle to be:
(x - 7)² + (y + 24)² = 625
Comparing this equation with the given options, we can see that option C matches our result:
C. (x - 7)² + (y + 24)² = 625
Therefore, option C is the correct answer. This process of comparing the derived equation with the given options is a crucial step in problem-solving. It ensures that we have correctly applied the concepts and formulas and that our final answer matches the expected format. In multiple-choice questions, this step also allows us to eliminate incorrect options and narrow down the possibilities, increasing the chances of selecting the correct answer. Understanding the structure and components of the equation of a circle is key to accurately identifying the correct option. The signs, coefficients, and constants in the equation all play a significant role in defining the circle's characteristics.
Conclusion
In conclusion, we successfully identified the equation of the circle with its center at (7, -24) and passing through the origin. We achieved this by understanding the standard equation of a circle, calculating the radius using the distance formula, and substituting the values into the equation. This exercise demonstrates the importance of understanding fundamental concepts in mathematics and applying them systematically to solve problems. The ability to work with equations of circles is a valuable skill in various areas of mathematics and its applications.
We began by reviewing the standard equation of a circle, which provides a framework for understanding the relationship between a circle's center, radius, and the coordinates of points on its circumference. We then used the distance formula to calculate the radius of the circle, given its center and a point it passes through. This step highlighted the connection between the Pythagorean theorem and the distance formula. Finally, we substituted the calculated radius and the given center coordinates into the standard equation of a circle to obtain the equation that represents the specific circle in this problem.
By carefully following these steps, we were able to confidently identify the correct option from the given choices. This problem serves as a good example of how a solid understanding of fundamental concepts and a systematic approach to problem-solving can lead to accurate solutions in mathematics.
