Which Equation Represents A Circle With Center (2,-8) And Radius 11

Understanding the equation of a circle is a fundamental concept in coordinate geometry. This article delves into the specifics of identifying the correct equation for a circle given its center and radius. Specifically, we address the question: Which equation represents a circle with a center at $(2, -8)$ and a radius of 11? We will explore the standard form of a circle's equation, dissect the given options, and clarify the reasoning behind the correct answer. This comprehensive explanation will solidify your understanding of circles and their equations, ensuring you can tackle similar problems with confidence.

Understanding the Standard Form of a Circle Equation

The standard form of a circle's equation is expressed as: $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ represents the coordinates of the center of the circle, and $r$ is the radius. This equation is derived from the Pythagorean theorem and provides a straightforward way to define a circle's properties on a coordinate plane. Grasping this form is crucial for both writing and interpreting circle equations. The values of h and k determine the circle’s position on the coordinate plane, while r dictates its size. Remember, the equation uses $(x - h)$ and $(y - k)$, so the signs of the center coordinates are reversed when inserted into the equation. This is a common area for mistakes, so paying close attention to the signs is essential for accuracy.

The standard form elegantly captures the geometric properties of a circle, making it a powerful tool in analytical geometry. By understanding this form, we can easily determine the center and radius of any circle given its equation, and conversely, we can write the equation of a circle if we know its center and radius. This ability is fundamental for solving a wide range of problems, from simple geometric constructions to more complex analytical challenges. Furthermore, the standard form allows us to quickly visualize the circle on a coordinate plane, enhancing our geometric intuition and problem-solving skills. For example, a circle with the equation $(x - 3)^2 + (y + 2)^2 = 16$ immediately tells us that the center is at $(3, -2)$ and the radius is $\sqrt{16} = 4$. This immediate translation from equation to geometric properties highlights the importance of mastering the standard form.

Knowing the standard form is also invaluable when dealing with transformations of circles. If a circle is translated, the values of h and k will change, reflecting the shift in the center’s coordinates. Similarly, if a circle’s size changes, the radius r will be affected. However, the fundamental structure of the equation $(x - h)^2 + (y - k)^2 = r^2$ remains the same, providing a consistent framework for analyzing these transformations. This consistency makes the standard form a cornerstone of circle geometry, allowing us to predict and understand how changes to the circle’s properties are reflected in its equation. In essence, the standard form is more than just a formula; it is a bridge connecting the algebraic representation of a circle with its geometric reality.

Analyzing the Given Information

In this specific problem, we are given that the center of the circle is at $(2, -8)$ and the radius is 11. This information is crucial for constructing the correct equation. We know that the center coordinates will correspond to the $(h, k)$ values in the standard equation, and the radius will be used to calculate $r^2$. Substituting these values, we have $h = 2$, $k = -8$, and $r = 11$. This initial step of correctly identifying the values is critical for avoiding errors. Many students make mistakes by either misinterpreting the coordinates or forgetting to square the radius. Therefore, it's always beneficial to double-check these initial substitutions before proceeding further. By systematically breaking down the given information, we lay a solid foundation for accurately applying the standard form of the circle equation.

Applying these values to the standard form, we get $(x - 2)^2 + (y - (-8))^2 = 11^2$. Simplifying this, we have $(x - 2)^2 + (y + 8)^2 = 121$. This simplified equation is the direct result of substituting the given center and radius into the standard form. It’s important to note how the negative coordinate of the center, $-8$, becomes $+8$ in the equation because of the $(y - k)$ term. This sign change is a common source of errors, so it’s crucial to pay close attention to it. Furthermore, squaring the radius, 11, to get 121 is another critical step that must not be overlooked. The resulting equation, $(x - 2)^2 + (y + 8)^2 = 121$, represents all the points $(x, y)$ that lie on the circle with the specified center and radius. It’s a precise algebraic representation of the geometric figure, and understanding how to derive it is a key skill in coordinate geometry.

The process of substituting values into the standard form and simplifying the equation highlights the interconnectedness of algebra and geometry. The equation is not just an abstract formula; it is a concise way to describe a geometric object. By manipulating the equation, we can gain insights into the circle’s properties, such as its position and size. This ability to move between the algebraic and geometric representations is a hallmark of mathematical proficiency. For instance, if we were to graph this circle, we could easily verify that the center is indeed at $(2, -8)$ and that the distance from the center to any point on the circle is 11 units. This visual confirmation further solidifies our understanding of the equation and its geometric interpretation.

Evaluating the Answer Choices

Now, let's evaluate the provided answer choices based on our understanding of the standard form and the given information. We are looking for an equation that matches $(x - 2)^2 + (y + 8)^2 = 121$. Each option presents a slightly different variation, and it's essential to systematically compare them to our derived equation. This process involves carefully examining the signs and values in each term to identify the correct match. This comparative analysis not only helps in finding the right answer but also reinforces our understanding of how the different components of the equation correspond to the circle’s characteristics.

• Option A: $(x - 8)^2 + (y + 2)^2 = 11$

  This option has the center coordinates reversed and uses the radius instead of the radius squared. The x-term suggests a center x-coordinate of 8, and the y-term suggests a center y-coordinate of -2. Furthermore, the right-hand side of the equation is 11, indicating a radius of $\sqrt{11}$, not 11. Therefore, this option is incorrect.

• Option B: $(x - 2)^2 + (y + 8)^2 = 121$

  This option perfectly matches our derived equation. The x-term $(x - 2)^2$ correctly represents the center's x-coordinate of 2, and the y-term $(y + 8)^2$ correctly represents the center's y-coordinate of -8. The right-hand side, 121, is the square of the radius, 11. This option accurately represents a circle with a center at $(2, -8)$ and a radius of 11, making it the correct answer.

• Option C: $(x + 2)^2 + (y - 8)^2 = 11$

  This option has the signs of the center coordinates reversed and uses the radius instead of the radius squared. The x-term suggests a center x-coordinate of -2, and the y-term suggests a center y-coordinate of 8. The right-hand side of the equation is 11, indicating a radius of $\sqrt{11}$, not 11. Thus, this option is incorrect.

• Option D: $(x + 8)^2 + (y - 2)^2 = 121$

  This option has the center coordinates significantly altered. The x-term $(x + 8)^2$ suggests a center x-coordinate of -8, and the y-term $(y - 2)^2$ suggests a center y-coordinate of 2. While the right-hand side, 121, is the correct square of the radius, the incorrect center coordinates disqualify this option.

Through this systematic evaluation, we can confidently identify Option B as the correct equation.

Conclusion

In conclusion, the correct equation representing a circle with a center at $(2, -8)$ and a radius of 11 is $(x - 2)^2 + (y + 8)^2 = 121$. This determination was made by understanding the standard form of a circle's equation, correctly substituting the given values, and systematically evaluating the answer choices. Mastering the standard form of a circle's equation is crucial for solving problems in coordinate geometry. The ability to accurately translate between the geometric properties of a circle and its algebraic representation is a fundamental skill in mathematics. This understanding allows us to confidently tackle a wide range of problems involving circles and their equations. By carefully applying the principles discussed in this article, you can confidently solve similar problems and enhance your understanding of circles in coordinate geometry.

Circle Equation Explained Find Equation for Center (2,-8) Radius 11