What Is The Center Of A Circle Represented By The Equation \[$(x+9)^2+(y-6)^2=10^2\$\]?A. \[$(-9, 6)\$\] B. \[$(-6, 9)\$\] C. \[$(6, -9)\$\] D. \[$(9, -6)\$\]

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The equation of a circle in standard form is given by (x−h)2+(y−k)2=r2{(x-h)^2+(y-k)^2=r^2}, where (h,k){(h,k)} represents the center of the circle and r{r} is the radius. In the given equation {(x+9)2+(y-6)2=10^2$}$, we need to identify the center of the circle.

Breaking Down the Equation

To find the center of the circle, we need to look at the terms inside the parentheses. The equation can be rewritten as (x+9)2+(y−6)2=100{(x+9)^2+(y-6)^2=100}. The terms x+9{x+9} and y−6{y-6} indicate that the center of the circle is being translated 9{9} units to the left and 6{6} units up from the origin.

Identifying the Center

The center of the circle is represented by the point (h,k){(h,k)}. In this case, h=−9{h=-9} and k=6{k=6}. Therefore, the center of the circle is (−9,6){(-9,6)}.

Analyzing the Options

Let's analyze the options given:

  • A. (−9,6){(-9,6)}
  • B. (−6,9){(-6,9)}
  • C. (6,−9){(6,-9)}
  • D. (9,−6){(9,-6)}

Only option A matches the coordinates we found for the center of the circle.

Conclusion

In conclusion, the center of the circle represented by the equation {(x+9)2+(y-6)2=10^2$}$ is (−9,6){(-9,6)}. This is the correct answer.

Key Takeaways

  • The equation of a circle in standard form is given by (x−h)2+(y−k)2=r2{(x-h)^2+(y-k)^2=r^2}.
  • The center of the circle is represented by the point (h,k){(h,k)}.
  • To find the center of the circle, look at the terms inside the parentheses in the equation.
  • The center of the circle is (−9,6){(-9,6)} in the given equation.

Practice Problems

  1. Find the center of the circle represented by the equation {(x-4)2+(y+3)2=16$}$.
  2. Find the center of the circle represented by the equation {(x+2)2+(y-5)2=9$}$.

Solutions

  1. The center of the circle is (4,−3){(4,-3)}.
  2. The center of the circle is (−2,5){(-2,5)}.

Additional Resources

For more information on the equation of a circle and how to find the center, check out the following resources:

  • Khan Academy: Equation of a Circle
  • Mathway: Equation of a Circle
  • Wolfram Alpha: Equation of a Circle

Final Thoughts

Q: What is the standard form of the equation of a circle?

A: The standard form of the 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){(h,k)} represents the center of the circle and r{r} is the radius.

Q: How do I find the center of a circle represented by an equation?

A: To find the center of a circle represented by an equation, look at the terms inside the parentheses. The center of the circle is represented by the point (h,k){(h,k)}. For example, in the equation {(x+9)2+(y-6)2=10^2$}$, the center of the circle is (−9,6){(-9,6)}.

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

A: The radius of a circle is the distance from the center of the circle to any point on the circle. In the equation of a circle, the radius is represented by r{r}. The radius is used to determine the size of the circle.

Q: Can I find the center of a circle if I know the coordinates of a point on the circle?

A: Yes, you can find the center of a circle if you know the coordinates of a point on the circle. To do this, you need to use the equation of the circle and substitute the coordinates of the point into the equation. Then, solve for the center of the circle.

Q: How do I determine the radius of a circle represented by an equation?

A: To determine the radius of a circle represented by an equation, look at the right-hand side of the equation. The radius is the square root of the number on the right-hand side. For example, in the equation {(x+9)2+(y-6)2=10^2$}$, the radius is 10{10}.

Q: Can I find the equation of a circle if I know the center and radius?

A: Yes, you can find the equation of a circle if you know the center and radius. To do this, use the standard form of the equation of a circle and substitute the values of the center and radius into the equation.

Q: What are some common mistakes to avoid when working with the equation of a circle?

A: Some common mistakes to avoid when working with the equation of a circle include:

  • Not using the standard form of the equation of a circle.
  • Not identifying the center of the circle correctly.
  • Not determining the radius correctly.
  • Not using the correct values for the center and radius when finding the equation of a circle.

Q: How can I practice working with the equation of a circle?

A: You can practice working with the equation of a circle by:

  • Solving problems and exercises that involve the equation of a circle.
  • Using online resources and tools to practice working with the equation of a circle.
  • Working with a tutor or teacher to get help and feedback on your work.

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

A: Some real-world applications of the equation of a circle include:

  • Designing and building circular structures such as bridges, tunnels, and buildings.
  • Creating and analyzing circular patterns and shapes in art and design.
  • Working with circular motion and rotation in physics and engineering.
  • Using the equation of a circle to model and analyze real-world phenomena such as the motion of planets and stars.

Conclusion

The equation of a circle is a fundamental concept in mathematics that has many real-world applications. By understanding the standard form of the equation of a circle and how to find the center and radius, you can solve problems and exercises that involve the equation of a circle. Practice working with the equation of a circle by solving problems and exercises, using online resources and tools, and working with a tutor or teacher.