Select The Correct Answer.Which Equation Represents A Circle With A Center At \[$(-5, 5)\$\] And A Radius Of 3 Units?A. \[$(x+5)^2+(y-5)^2=6\$\]B. \[$(x-5)^2+(y+5)^2=3\$\]C. \[$(x-5)^2+(y+5)^2=9\$\]D.

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Introduction

Circles are a fundamental concept in mathematics, and understanding how to represent them algebraically is crucial for solving various problems in geometry, trigonometry, and other branches of mathematics. In this article, we will delve into the world of circles and explore how to select the correct equation that represents a circle with a given center and radius.

What is a Circle?

A circle is a set of points in a plane that are equidistant from a fixed point, known as the center. The distance between the center and any point on the circle is called the radius. Circles can be represented graphically using a circle symbol or algebraically using an equation.

Standard Equation of a Circle

The standard equation of a circle with center (h,k){(h, k)} and radius r{r} is given by:

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

Selecting the Correct Equation

Now that we have the standard equation of a circle, let's apply it to the given problem. We are asked to select the correct equation that represents a circle with a center at (−5,5){(-5, 5)} and a radius of 3 units.

Step 1: Identify the Center and Radius

The center of the circle is given as (−5,5){(-5, 5)}, and the radius is 3 units.

Step 2: Plug in the Values into the Standard Equation

Substitute the values of the center and radius into the standard equation:

(x−(−5))2+(y−5)2=32{(x - (-5))^2 + (y - 5)^2 = 3^2}

Simplify the equation:

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

Step 3: Compare with the Options

Now, let's compare the derived equation with the given options:

A. (x+5)2+(y−5)2=6{(x+5)^2+(y-5)^2=6} B. (x−5)2+(y+5)2=3{(x-5)^2+(y+5)^2=3} C. (x−5)2+(y+5)2=9{(x-5)^2+(y+5)^2=9} D. (x+5)2+(y+5)2=9{(x+5)^2+(y+5)^2=9}

Conclusion

Based on the derived equation, the correct answer is:

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){(-5, 5)} and a radius of 3 units.

Why is this the Correct Answer?

The correct answer is C because it matches the derived equation exactly. The equation has the correct center and radius, and the squared terms are correctly expanded.

Tips and Tricks

When selecting the correct equation, make sure to:

  • Identify the center and radius of the circle
  • Plug in the values into the standard equation
  • Simplify the equation
  • Compare the derived equation with the given options

By following these steps, you can confidently select the correct equation that represents a circle with a given center and radius.

Real-World Applications

Understanding circles and their equations has numerous real-world applications, including:

  • Geometry and Trigonometry: Circles are used to represent geometric shapes and trigonometric functions.
  • Physics and Engineering: Circles are used to model real-world phenomena, such as the motion of objects and the design of mechanical systems.
  • Computer Science: Circles are used in computer graphics and game development to create realistic and engaging visual effects.

Conclusion

Introduction

In our previous article, we explored the standard equation of a circle and how to select the correct equation that represents a circle with a given center and radius. In this article, we will continue to delve into the world of circle equations and provide a Q&A guide to help you better understand and apply these concepts.

Q: What is the standard equation of a circle?

A: The standard equation of a circle with center (h,k){(h, k)} and radius r{r} is given by:

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

Q: How do I select the correct equation that represents a circle with a given center and radius?

A: To select the correct equation, follow these steps:

  1. Identify the center and radius of the circle
  2. Plug in the values into the standard equation
  3. Simplify the equation
  4. Compare the derived equation with the given options

Q: What is the difference between the standard equation and the general equation of a circle?

A: The standard equation of a circle is a specific form of the general equation, which is:

Ax2+Bxy+Cy2+Dx+Ey+F=0{Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0}

The standard equation is a special case of the general equation where A=C=1{A = C = 1}, B=0{B = 0}, and D=E=−2h{D = E = -2h} and F=k2−r2{F = k^2 - r^2}.

Q: How do I convert the general equation of a circle to the standard equation?

A: To convert the general equation to the standard equation, follow these steps:

  1. Complete the square for the x{x} and y{y} terms
  2. Simplify the equation
  3. Compare the derived equation with the standard equation

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

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

Q: How do I find the center and radius of a circle from its equation?

A: To find the center and radius of a circle from its equation, follow these steps:

  1. Identify the values of h{h} and k{k} in the equation
  2. Calculate the value of r{r} using the equation r2=k2−h2{r^2 = k^2 - h^2}

Q: What are some common applications of circle equations in real-world problems?

A: Circle equations have numerous real-world applications, including:

  • Geometry and Trigonometry: Circles are used to represent geometric shapes and trigonometric functions.
  • Physics and Engineering: Circles are used to model real-world phenomena, such as the motion of objects and the design of mechanical systems.
  • Computer Science: Circles are used in computer graphics and game development to create realistic and engaging visual effects.

Q: How do I use circle equations to solve problems in geometry and trigonometry?

A: To use circle equations to solve problems in geometry and trigonometry, follow these steps:

  1. Identify the problem and the relevant circle equation
  2. Apply the equation to the problem
  3. Solve for the unknown values

Conclusion

In conclusion, circle equations are a fundamental concept in mathematics, and understanding how to select the correct equation and apply it to real-world problems is crucial for success in geometry, trigonometry, and other branches of mathematics. By following the steps outlined in this Q&A guide, you can confidently apply circle equations to a variety of mathematical and real-world problems.

Additional Resources

For further learning and practice, we recommend the following resources:

  • Math textbooks: "Geometry: Seeing, Doing, Understanding" by Harold R. Jacobs and "Trigonometry: A Unit Circle Approach" by Michael Corral
  • Online resources: Khan Academy, MIT OpenCourseWare, and Wolfram Alpha
  • Practice problems: Circle equations practice problems on Brilliant, Mathway, and Symbolab

Final Thoughts

Circle equations are a powerful tool for solving problems in geometry, trigonometry, and other branches of mathematics. By mastering the standard equation and applying it to real-world problems, you can unlock new insights and solutions to complex problems. Remember to practice regularly and seek help when needed to become proficient in using circle equations.