Select The Correct Answer.Which Equation Represents A Circle With A Center At { (-5, 5)$}$ And A Radius Of 3 Units?A. { (x+5)^2 + (y-5)^2 = 3$}$ B. { (x-5)^2 + (y+5)^2 = 3$}$ C. { (x+5)^2 + (y-5)^2 = 9$}$ D.

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A circle is a set of points that are all equidistant from a central point, known as the center. The equation of a circle is a fundamental concept in mathematics, and it is used to represent the circle on a coordinate plane. In this article, we will explore the equation of a circle and how to select the correct answer when given a set of options.

The General Equation of a Circle

The general equation of a circle is given by:

(x−h)2+(y−k)2=r2{(x-h)^2 + (y-k)^2 = r^2}

where (h, k) is the center of the circle and r is the radius. This equation represents a circle with a center at (h, k) and a radius of r units.

Selecting the Correct Answer

Given the equation of a circle, we need to select the correct answer from a set of options. In this case, we are given a circle with a center at (-5, 5) and a radius of 3 units. We need to select the equation that represents this circle.

Option A: (x+5)2+(y−5)2=3{(x+5)^2 + (y-5)^2 = 3}

This option is incorrect because the radius is squared, which means it should be equal to 9, not 3.

Option B: (x−5)2+(y+5)2=3{(x-5)^2 + (y+5)^2 = 3}

This option is also incorrect because the center is not at (-5, 5), but rather at (5, -5).

Option C: (x+5)2+(y−5)2=9{(x+5)^2 + (y-5)^2 = 9}

This option is correct because the center is at (-5, 5) and the radius is squared, which means it is equal to 9.

Conclusion

In conclusion, the correct answer is Option C: (x+5)2+(y−5)2=9{(x+5)^2 + (y-5)^2 = 9}. This equation represents a circle with a center at (-5, 5) and a radius of 3 units.

Understanding the Importance of the Center and Radius

The center and radius of a circle are crucial components of its equation. The center represents the point from which all points on the circle are equidistant, while the radius represents the distance from the center to any point on the circle. Understanding the importance of the center and radius is essential in selecting the correct answer when given a set of options.

Real-World Applications of the Equation of a Circle

The equation of a circle has numerous real-world applications, including:

  • Geometry and Trigonometry: The equation of a circle is used to represent circles on a coordinate plane, which is essential in geometry and trigonometry.
  • Physics and Engineering: The equation of a circle is used to represent the path of an object in motion, which is essential in physics and engineering.
  • Computer Science: The equation of a circle is used in computer graphics and game development to represent circles and other shapes.

Common Mistakes to Avoid

When selecting the correct answer, there are several common mistakes to avoid, including:

  • Not squaring the radius: The radius should be squared, which means it should be equal to the square of the radius.
  • Not using the correct center: The center should be at the correct coordinates, which means it should be at (h, k).
  • Not using the correct equation: The equation should be in the correct form, which means it should be in the form (x-h)^2 + (y-k)^2 = r^2.

Conclusion

In our previous article, we explored the equation of a circle and how to select the correct answer when given a set of options. In this article, we will answer some frequently asked questions about the equation of a circle.

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=r2{(x-h)^2 + (y-k)^2 = r^2}

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

Q: What is the significance of the center and radius in the equation of a circle?

A: The center and radius are crucial components of the equation of a circle. The center represents the point from which all points on the circle are equidistant, while the radius represents the distance from the center to any point on the circle.

Q: How do I select the correct answer when given a set of options?

A: To select the correct answer, you need to ensure that the center and radius are correct. The center should be at the correct coordinates, and the radius should be squared.

Q: What are some common mistakes to avoid when selecting the correct answer?

A: Some common mistakes to avoid include:

  • Not squaring the radius
  • Not using the correct center
  • Not using the correct equation

Q: What are some real-world applications of the equation of a circle?

A: The equation of a circle has numerous real-world applications, including:

  • Geometry and Trigonometry
  • Physics and Engineering
  • Computer Science

Q: How do I use the equation of a circle in geometry and trigonometry?

A: In geometry and trigonometry, the equation of a circle is used to represent circles on a coordinate plane. This is essential in understanding the properties of circles and how they relate to other shapes.

Q: How do I use the equation of a circle in physics and engineering?

A: In physics and engineering, the equation of a circle is used to represent the path of an object in motion. This is essential in understanding the motion of objects and how to design systems that can handle circular motion.

Q: How do I use the equation of a circle in computer science?

A: In computer science, the equation of a circle is used in computer graphics and game development to represent circles and other shapes. This is essential in creating realistic and engaging graphics and games.

Q: What are some tips for memorizing the equation of a circle?

A: Some tips for memorizing the equation of a circle include:

  • Writing it down multiple times
  • Creating flashcards
  • Practicing problems that involve the equation of a circle

Q: What are some resources for learning more about the equation of a circle?

A: Some resources for learning more about the equation of a circle include:

  • Online tutorials and videos
  • Textbooks and workbooks
  • Online communities and forums

Conclusion

In conclusion, the equation of a circle is a fundamental concept in mathematics, and it is used to represent circles on a coordinate plane. By understanding the equation of a circle and how to select the correct answer, you can apply it to a variety of real-world applications. Remember to avoid common mistakes and use the equation of a circle in geometry, trigonometry, physics, engineering, and computer science.