Equation Of A Circle Passing Through A Point And Having A Center

Determining the equation of a circle given a point and its center is a fundamental concept in coordinate geometry. This article delves into the process of finding the correct equation, using the distance formula and the standard form of a circle's equation. We will explore the underlying principles, step-by-step calculations, and common pitfalls to avoid. This comprehensive guide will not only provide the solution to the specific problem but also equip you with the knowledge to tackle similar questions with confidence.

Understanding the Circle Equation

To understand the circle equation, it's crucial to grasp the standard form: $(x - h)^2 + (y - k)^2 = r^2$. Here, $(h, k)$ represents the center of the circle, and $r$ denotes its radius. This equation is derived from the Pythagorean theorem and the distance formula, which calculates the distance between two points in a coordinate plane. The radius, in this context, is the constant distance between the center of the circle and any point on its circumference. When we're given the center and a point on the circle, we can utilize the distance formula to determine the radius. This process involves substituting the coordinates of the center and the given point into the distance formula and solving for the distance, which represents the radius. Once we have the radius, we can square it to find $r^2$, which is the value we need for the standard form of the circle equation. By understanding these foundational concepts, we can confidently approach problems involving circles and their equations, ensuring accurate and efficient solutions. Furthermore, recognizing the relationship between the equation and the geometric properties of the circle allows us to visualize and interpret the equation more effectively. This holistic understanding is invaluable for both solving problems and gaining a deeper appreciation of coordinate geometry.

Applying the Distance Formula

Applying the distance formula is the key to finding the radius of the circle. The distance formula, expressed as $\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}$, calculates the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ in a coordinate plane. In our scenario, the two points are the center of the circle $(4, 0)$ and the point on the circle $(-2, 8)$. We substitute these coordinates into the formula: $\sqrt{\left(-2-4\right)^2+\left(8-0\right)^2}$. Simplifying the expression inside the parentheses, we get $\sqrt{\left(-6\right)^2+\left(8\right)^2}$. Squaring the numbers, we have $\sqrt{36+64}$, which further simplifies to $\sqrt{100}$. Therefore, the radius $r$ is $\sqrt{100} = 10$. This radius represents the distance between the center of the circle and the given point, and it is a crucial component in determining the equation of the circle. Understanding how to apply the distance formula correctly is essential not only for this specific problem but also for various other geometric problems involving distances and coordinates. The formula provides a direct and accurate method for calculating the length of a line segment in a coordinate plane, making it a fundamental tool in coordinate geometry. By mastering this formula, you can confidently solve problems related to circles, triangles, and other geometric shapes.

Constructing the Circle Equation

After constructing the circle equation after determining the radius, we can now plug the values into the standard form of a circle's equation: $(x - h)^2 + (y - k)^2 = r^2$. We know the center of the circle is $(4, 0)$, so $h = 4$ and $k = 0$. We also calculated the radius to be $r = 10$, so $r^2 = 100$. Substituting these values into the standard form, we get $(x - 4)^2 + (y - 0)^2 = 100$. Simplifying, the equation becomes $(x - 4)^2 + y^2 = 100$. This equation represents the circle that contains the point $(-2, 8)$ and has a center at $(4, 0)$. It's essential to remember that the signs in the equation are opposite to the coordinates of the center. For example, since the center's x-coordinate is 4, the term in the equation is $(x - 4)$. Similarly, since the center's y-coordinate is 0, the term is $(y - 0)$, which simplifies to $y$. The right side of the equation is the square of the radius, so we use $r^2 = 100$. By understanding this process, you can confidently construct the equation of a circle given its center and radius, or given the center and a point on the circle. This skill is fundamental in coordinate geometry and is applicable to various problems involving circles and their properties.

Analyzing the Answer Choices

When analyzing the answer choices, it's crucial to compare them with the equation we derived: $(x - 4)^2 + y^2 = 100$. Option A, $(x - 4)^2 + y^2 = 100$, perfectly matches our derived equation. This confirms that option A is the correct answer. Let's examine the other options to understand why they are incorrect. Option B, $(x - 4)^2 + y^2 = 10$, has the correct center but an incorrect radius. The right side of the equation is 10, which means the radius would be $\sqrt{10}$, not 10. Option C, $x^2 + (y - 4)^2 = 10$, has an incorrect center and radius. The center would be $(0, 4)$, and the radius would be $\sqrt{10}$. Option D, $x^2 + (y - 4)^2 = 100$, has the correct radius but an incorrect center. The center would be $(0, 4)$. By carefully comparing each option with our derived equation and identifying the discrepancies, we can confidently eliminate the incorrect choices and select the correct answer. This process of elimination is a valuable strategy for solving multiple-choice questions, especially in mathematics. It allows you to focus on the key differences between the options and make an informed decision based on your calculations and understanding of the concepts.

Conclusion

In conclusion, the equation that represents a circle containing the point $(-2, 8)$ and having a center at $(4, 0)$ is $(x - 4)^2 + y^2 = 100$. This was determined by applying the distance formula to find the radius and then substituting the center coordinates and radius into the standard form of a circle's equation. Understanding the distance formula and the standard form of a circle's equation are fundamental concepts in coordinate geometry. The distance formula allows us to calculate the distance between two points, which in this case, helped us find the radius of the circle. The standard form of a circle's equation, $(x - h)^2 + (y - k)^2 = r^2$, provides a clear representation of the circle's center and radius. By mastering these concepts, you can confidently solve a wide range of problems involving circles and their equations. This problem-solving approach involves several key steps: first, understanding the problem and identifying the given information; second, applying relevant formulas and concepts to derive the required values; and third, constructing the equation based on these values. This methodical approach is applicable not only to this specific problem but also to various other mathematical problems. Remember to practice and reinforce these concepts to build your problem-solving skills and deepen your understanding of coordinate geometry.