Finding The Equation Of A Circle Center (4, 0) And Point (-2, 8)
Introduction
In this comprehensive article, we will delve into the problem of identifying the equation of a circle that passes through a specific point and has a defined center. This is a fundamental concept in analytic geometry, combining the geometric properties of circles with algebraic equations. Understanding how to determine the equation of a circle given its center and a point on its circumference is crucial for various applications in mathematics, physics, and engineering. We will meticulously walk through the process, breaking down each step and providing a clear explanation to ensure a thorough understanding. Let’s embark on this geometric journey to master the art of circle equations.
Understanding the Standard Equation of a Circle
The cornerstone of solving this problem lies in the standard equation of a circle. This equation provides a concise and powerful way to represent a circle in the Cartesian coordinate system. The standard equation is given by:
(x - h)² + (y - k)² = r²
Where:
- (h, k) represents the coordinates of the center of the circle.
- r denotes the radius of the circle, which is the distance from the center to any point on the circle's circumference.
- (x, y) represents any point on the circumference of the circle.
This equation beautifully encapsulates the geometric definition of a circle: the set of all points equidistant (the radius) from a fixed point (the center). By understanding and manipulating this equation, we can solve a variety of problems related to circles, including finding the equation of a circle given certain conditions.
Key Components of the Standard Equation
Let's break down the key components of the standard equation to gain a deeper understanding:
- (x - h)² and (y - k)²: These terms represent the squared horizontal and vertical distances, respectively, from any point (x, y) on the circle to the center (h, k). They are derived from the Pythagorean theorem, which relates the sides of a right triangle. In this context, the horizontal and vertical distances form the legs of the right triangle, and the radius is the hypotenuse.
- r²: This term represents the square of the radius. The radius, as mentioned earlier, is the constant distance from the center to any point on the circle. Squaring the radius in the equation ensures that we are dealing with distances in a consistent manner, aligning with the Pythagorean theorem.
By grasping the significance of each component, we can effectively utilize the standard equation to solve circle-related problems. We will see how this equation plays a pivotal role in determining the equation of the circle in our given problem.
Applying the Distance Formula
To determine the equation of the circle, we need to find the radius. The problem provides us with the center of the circle, (4, 0), and a point on the circle, (-2, 8). The distance formula comes into play here, as it allows us to calculate the distance between two points in a coordinate plane. This distance, in our case, will be the radius of the circle.
The distance formula is derived from the Pythagorean theorem and is given by:
√[(x₂ - x₁)² + (y₂ - y₁)²]
Where:
- (x₁, y₁) and (x₂, y₂) are the coordinates of the two points.
Calculating the Radius
Let’s apply the distance formula to find the radius of our circle. We have the center (4, 0) as one point and the point on the circle (-2, 8) as the other. Plugging these values into the distance formula, we get:
r = √[(-2 - 4)² + (8 - 0)²]
Let's simplify this step by step:
r = √[(-6)² + (8)²] r = √[36 + 64] r = √100 r = 10
Therefore, the radius of the circle is 10 units. This calculation is crucial because the radius is a key component of the circle's equation. Now that we have the radius, we can proceed to plug it into the standard equation of a circle.
Forming the Circle's Equation
Now that we have determined the radius (r = 10) and we know the center of the circle (h = 4, k = 0), we can construct the equation of the circle. We will use the standard equation of a circle:
(x - h)² + (y - k)² = r²
Substituting the values we have, we get:
(x - 4)² + (y - 0)² = 10²
Simplifying this, we have:
(x - 4)² + y² = 100
This is the equation of the circle that contains the point (-2, 8) and has its center at (4, 0). This equation represents all the points (x, y) that lie on the circumference of the circle. It is a powerful representation that encapsulates the geometric properties of the circle in algebraic form.
Verifying the Solution
To ensure that our equation is correct, we can substitute the coordinates of the given point (-2, 8) into the equation and check if it holds true. Plugging in x = -2 and y = 8 into our equation, we get:
((-2) - 4)² + (8)² = 100
Simplifying:
(-6)² + 64 = 100
36 + 64 = 100
100 = 100
Since the equation holds true, we can confidently say that the equation (x - 4)² + y² = 100 correctly represents the circle. This verification step is a crucial part of the problem-solving process, as it confirms the accuracy of our solution.
Analyzing the Answer Choices
Having derived the equation of the circle, let's now compare it to the given answer choices to identify the correct option. The answer choices provided are:
A. (x - 4)² + y² = 100 B. (x - 4)² + y² = 10 C. x² + (y - 4)² = 10 D. x² + (y - 4)² = 100
By comparing our derived equation, (x - 4)² + y² = 100, with the answer choices, it is clear that option A, (x - 4)² + y² = 100, is the correct answer. The other options differ in either the position of the center or the value of the radius, making them incorrect representations of the circle described in the problem.
Why Other Options Are Incorrect
To further solidify our understanding, let's analyze why the other options are incorrect:
- Option B: (x - 4)² + y² = 10: This equation represents a circle with the same center (4, 0) but with a radius of √10, not 10. Therefore, it does not pass through the point (-2, 8).
- Option C: x² + (y - 4)² = 10: This equation represents a circle with a center at (0, 4) and a radius of √10. It has neither the correct center nor the correct radius.
- Option D: x² + (y - 4)² = 100: This equation represents a circle with a center at (0, 4) and a radius of 10. While it has the correct radius, the center is incorrect.
Understanding why these options are incorrect reinforces our grasp of the standard equation of a circle and the importance of accurately identifying the center and radius.
Conclusion
In conclusion, the equation that represents a circle containing the point (-2, 8) and having a center at (4, 0) is (x - 4)² + y² = 100. We arrived at this solution by applying the distance formula to find the radius and then substituting the center coordinates and radius into the standard equation of a circle. We also verified our solution by plugging the given point into the equation and confirming that it holds true. Furthermore, we analyzed the other answer choices to understand why they were incorrect, reinforcing our understanding of circle equations.
Key Takeaways
- The standard equation of a circle is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius.
- The distance formula, √[(x₂ - x₁)² + (y₂ - y₁)²], is used to calculate the distance between two points, which can be used to find the radius of a circle.
- To find the equation of a circle, you need to know the center and the radius.
- Always verify your solution by plugging in given points into the equation.
This problem illustrates a fundamental concept in analytic geometry. Mastering these skills is essential for further studies in mathematics and related fields. By understanding the standard equation of a circle and how to apply the distance formula, you can confidently tackle similar problems and expand your mathematical toolkit. Remember to practice regularly and break down complex problems into smaller, manageable steps. With dedication and a solid understanding of the concepts, you can excel in solving circle-related problems and more!
