Equation Of A Circle With Center (2 -8) And Radius 11

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Understanding the equation of a circle is a fundamental concept in coordinate geometry. This article will thoroughly explore how to determine the equation of a circle given its center and radius. We will specifically focus on identifying the correct equation for a circle with a center at (2, -8) and a radius of 11. This exploration will not only help in solving this particular problem but also provide a comprehensive understanding of the general form of a circle's equation and its applications.

The standard form of the equation of a circle is a powerful tool that allows us to easily represent a circle on the coordinate plane. This form is expressed as:

(x - h)² + (y - k)² = r²

Where:

  • (h, k) represents the coordinates of the center of the circle.
  • r represents the radius of the circle.

This equation is derived from the Pythagorean theorem, which relates the sides of a right triangle. In the context of a circle, the distance from any point (x, y) on the circle to the center (h, k) is always equal to the radius r. This distance can be calculated using the distance formula, which is a direct application of the Pythagorean theorem. By squaring both sides of the distance formula, we arrive at the standard equation of a circle. Understanding the origin of this equation helps in remembering and applying it correctly.

To apply this formula, we need to identify the center (h, k) and the radius (r) from the given information. In our case, the center is given as (2, -8), and the radius is given as 11. Substituting these values into the standard equation of a circle, we get:

(x - 2)² + (y - (-8))² = 11²

Simplifying this equation, we have:

(x - 2)² + (y + 8)² = 121

This equation represents a circle with a center at (2, -8) and a radius of 11. The values of h and k determine the position of the center on the coordinate plane, while the value of r² determines the size of the circle. A larger value of r² indicates a larger radius and, consequently, a larger circle. Conversely, a smaller value of r² indicates a smaller radius and a smaller circle. The equation provides a concise way to describe a circle's geometric properties algebraically.

Now, let's examine the given options to identify the equation that matches our derived equation. This process will involve comparing the coefficients and constants in each option with the standard form of the circle equation. By carefully analyzing each option, we can pinpoint the correct equation that represents the circle with the specified center and radius. This step is crucial in ensuring that we have correctly applied the formula and understood the relationship between the equation and the circle's properties.

To find the correct equation, we need to compare the provided options with the standard form we derived: (x - 2)² + (y + 8)² = 121. Each option presents a slightly different equation, and we must carefully examine each one to determine which matches our derived equation.

Let's break down each option:

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

    This equation suggests a circle with a center at (8, -2) and a radius of √11. Notice that the signs inside the parentheses are opposite to the coordinates of the center. Also, the right side of the equation represents the square of the radius, so 11 implies a radius of √11, not 11. This option does not match our required center and radius.

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

    This equation perfectly matches our derived equation. It represents a circle with a center at (2, -8) and a radius of √121 = 11. The equation correctly incorporates the center coordinates with the appropriate signs and the squared radius value. This is the correct option.

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

    This equation represents a circle with a center at (-2, 8) and a radius of √11. The signs inside the parentheses indicate a center with coordinates that are the opposite of the constants within the parentheses. Additionally, the right side of the equation, 11, implies a radius of √11, not 11. This option does not match our required center and radius.

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

    This equation represents a circle with a center at (-8, 2) and a radius of √121 = 11. The signs inside the parentheses indicate a center with coordinates that are the opposite of the constants within the parentheses. While the radius is correct (√121 = 11), the center does not match our required center of (2, -8). Therefore, this option is incorrect.

Through this detailed analysis, we can confidently conclude that Option B is the only equation that accurately represents a circle with a center at (2, -8) and a radius of 11. The other options either have incorrect centers or incorrect radii, making them unsuitable for the given conditions. This process highlights the importance of carefully examining each component of the equation and comparing it to the given information.

This exercise reinforces several key takeaways regarding the equation of a circle. Firstly, understanding the standard form of the equation, (x - h)² + (y - k)² = r², is crucial. This form directly relates the center (h, k) and the radius (r) to the equation, making it easy to identify these properties or construct the equation if given the center and radius.

Secondly, paying close attention to the signs in the equation is essential. The coordinates of the center are represented as (h, k), but in the equation, they appear as (x - h) and (y - k). This means that if the center has a positive x-coordinate, it will appear as a negative value within the parentheses, and vice versa. Similarly, the y-coordinate follows the same rule. For instance, a center at (2, -8) translates to (x - 2) and (y + 8) in the equation.

Thirdly, remember that the value on the right side of the equation is the square of the radius (r²). Therefore, if you are given the equation and need to find the radius, you must take the square root of this value. Conversely, when constructing the equation, ensure you square the radius before placing it on the right side.

Finally, practice is key to mastering the circle equation. Working through various examples and problems helps solidify your understanding of the concepts and allows you to apply the equation confidently in different scenarios. This includes problems where you are given the center and radius, as well as problems where you are given the equation and need to identify the center and radius. By consistently practicing, you can develop a strong intuition for the circle equation and its applications.

In conclusion, the equation (x - 2)² + (y + 8)² = 121 represents a circle with a center at (2, -8) and a radius of 11. This determination was made by understanding the standard form of a circle's equation and carefully comparing it with the provided options. By paying attention to the signs and the squared radius, we were able to accurately identify the correct equation. This exercise serves as a valuable lesson in applying geometric concepts to algebraic equations and highlights the importance of careful analysis and attention to detail in mathematics.