Which Equation Represents A Circle With A Center At { (2, -8)$}$ And A Radius Of 11?A. { (x - 8)^2 + (y + 2)^2 = 11$}$B. { (x - 2)^2 + (y + 8)^2 = 121$}$C. { (x + 2)^2 + (y - 8)^2 = 11$} D . \[ D. \[ D . \[ (x + 8)^2 + (y

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Which Equation Represents a Circle with a Center at (2, -8) and a Radius of 11?

Understanding the Basics of a Circle Equation

A circle is a set of points that are all equidistant from a central point, known as the center. The distance from the center to any point on the circle is called the radius. In this article, we will explore how to write an equation that represents a circle with a given center and radius.

The General Equation of a Circle

The general equation of a circle is given by:

(x - h)^2 + (y - k)^2 = r^2

where (h, k) is the center of the circle and r is the radius.

Given Information

We are given that the center of the circle is at (2, -8) and the radius is 11. We need to find the equation that represents this circle.

Substituting the Given Values

To find the equation, we substitute the given values into the general equation:

(x - 2)^2 + (y - (-8))^2 = 11^2

Simplifying the Equation

We can simplify the equation by evaluating the expressions inside the parentheses:

(x - 2)^2 + (y + 8)^2 = 121

Comparing with the Options

Now, we compare the simplified equation with the given options:

A. (x - 8)^2 + (y + 2)^2 = 11 B. (x - 2)^2 + (y + 8)^2 = 121 C. (x + 2)^2 + (y - 8)^2 = 11 D. (x + 8)^2 + (y - 2)^2 = 121

Identifying the Correct Option

Based on the simplified equation, we can see that option B matches the equation:

(x - 2)^2 + (y + 8)^2 = 121

Conclusion

Therefore, the equation that represents a circle with a center at (2, -8) and a radius of 11 is:

(x - 2)^2 + (y + 8)^2 = 121

Key Takeaways

  • The general equation of a circle is given by (x - h)^2 + (y - k)^2 = r^2.
  • To find the equation of a circle with a given center and radius, substitute the values into the general equation.
  • Simplify the equation by evaluating the expressions inside the parentheses.
  • Compare the simplified equation with the given options to identify the correct one.

Practice Problems

Try the following problems to practice writing the equation of a circle:

  1. Find the equation of a circle with a center at (3, 4) and a radius of 5.
  2. Find the equation of a circle with a center at (-2, 1) and a radius of 3.
  3. Find the equation of a circle with a center at (0, 0) and a radius of 2.

Answer Key

  1. (x - 3)^2 + (y - 4)^2 = 25
  2. (x + 2)^2 + (y - 1)^2 = 9
  3. x^2 + y^2 = 4
    Circle Equation Q&A

Frequently Asked Questions about Circle Equations

In this article, we will answer some of the most frequently asked questions about circle equations. Whether you are a student, a teacher, or just someone who wants to learn more about circle equations, this article is for you.

Q: What is the general equation of a circle?

A: The general equation of a circle is given by:

(x - h)^2 + (y - k)^2 = r^2

where (h, k) is the center of the circle and r is the radius.

Q: How do I find the equation of a circle with a given center and radius?

A: To find the equation of a circle with a given center and radius, substitute the values into the general equation:

(x - h)^2 + (y - k)^2 = r^2

Q: What if the center of the circle is not at the origin (0, 0)?

A: If the center of the circle is not at the origin (0, 0), you need to substitute the given values into the general equation. For example, if the center is at (2, -8) and the radius is 11, the equation would be:

(x - 2)^2 + (y - (-8))^2 = 11^2

Q: How do I simplify the equation?

A: To simplify the equation, evaluate the expressions inside the parentheses. For example, in the equation above, (y - (-8))^2 simplifies to (y + 8)^2.

Q: What if I have a negative radius?

A: If you have a negative radius, it means that the circle is not a real circle. A circle must have a non-negative radius.

Q: Can I have a circle with a radius of 0?

A: Yes, you can have a circle with a radius of 0. This is called a degenerate circle, and it is a single point.

Q: How do I find the equation of a circle with a given diameter?

A: To find the equation of a circle with a given diameter, you need to find the radius first. The radius is half of the diameter. Then, you can use the general equation to find the equation of the circle.

Q: Can I have a circle with a diameter of 0?

A: No, you cannot have a circle with a diameter of 0. A circle must have a non-zero diameter.

Q: How do I find the equation of a circle with a given circumference?

A: To find the equation of a circle with a given circumference, you need to find the radius first. The circumference is given by C = 2Ï€r, where C is the circumference and r is the radius. Then, you can use the general equation to find the equation of the circle.

Q: Can I have a circle with a circumference of 0?

A: No, you cannot have a circle with a circumference of 0. A circle must have a non-zero circumference.

Conclusion

In this article, we have answered some of the most frequently asked questions about circle equations. Whether you are a student, a teacher, or just someone who wants to learn more about circle equations, this article is for you. We hope that you have found the information helpful and that you will be able to use it to solve problems involving circle equations.

Practice Problems

Try the following problems to practice writing the equation of a circle:

  1. Find the equation of a circle with a center at (3, 4) and a radius of 5.
  2. Find the equation of a circle with a center at (-2, 1) and a radius of 3.
  3. Find the equation of a circle with a center at (0, 0) and a radius of 2.
  4. Find the equation of a circle with a center at (2, -8) and a radius of 11.
  5. Find the equation of a circle with a center at (-5, 2) and a radius of 4.

Answer Key

  1. (x - 3)^2 + (y - 4)^2 = 25
  2. (x + 2)^2 + (y - 1)^2 = 9
  3. x^2 + y^2 = 4
  4. (x - 2)^2 + (y + 8)^2 = 121
  5. (x + 5)^2 + (y - 2)^2 = 16