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

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Understanding the Equation of a Circle

The equation of a circle in standard form is given by (x-h)^2 + (y-k)^2 = r^2, where (h,k) represents the coordinates of the center of the circle, and r is the radius of the circle. In the given equation (x+9)^2 + (y-6)^2 = 10^2, we can identify the values of h, k, and r.

Identifying the Center of the Circle

To find the center of the circle, we need to identify the values of h and k in the equation. By comparing the given equation with the standard form of the equation of a circle, we can see that h = -9 and k = 6.

Calculating the Center of the Circle

The center of the circle is represented by the coordinates (h,k), which in this case is (-9,6). Therefore, the correct answer is A. (-9,6).

Eliminating Incorrect Options

To confirm that the correct answer is A. (-9,6), let's eliminate the other options by substituting their coordinates into the equation.

Option B: (-6,9)

Substituting x = -6 and y = 9 into the equation, we get:

((-6)+9)^2 + ((9)-6)^2 = 10^2 (3)^2 + (3)^2 = 10^2 9 + 9 = 100 18 ≠ 100

Since the equation does not hold true for option B, we can eliminate it.

Option C: (6,-9)

Substituting x = 6 and y = -9 into the equation, we get:

((6)+9)^2 + ((-9)-6)^2 = 10^2 (15)^2 + (-15)^2 = 10^2 225 + 225 = 100 450 ≠ 100

Since the equation does not hold true for option C, we can eliminate it.

Option D: (9,-6)

Substituting x = 9 and y = -6 into the equation, we get:

((9)+9)^2 + ((-6)-6)^2 = 10^2 (18)^2 + (-12)^2 = 10^2 324 + 144 = 100 468 ≠ 100

Since the equation does not hold true for option D, we can eliminate it.

Conclusion

Based on the analysis, we can conclude that the center of the circle represented by the equation (x+9)^2 + (y-6)^2 = 10^2 is A. (-9,6).

Frequently Asked Questions

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

A: The equation of a circle in standard form is given by (x-h)^2 + (y-k)^2 = r^2, where (h,k) represents the coordinates of the center of the circle, and r is the radius of the circle.

Q: How do I identify the center of a circle from its equation?

A: To identify the center of a circle from its equation, you need to compare the given equation with the standard form of the equation of a circle and identify the values of h and k.

Q: What is the radius of the circle represented by the equation (x+9)^2 + (y-6)^2 = 10^2?

A: The radius of the circle represented by the equation (x+9)^2 + (y-6)^2 = 10^2 is 10.

References

Related Topics

  • [1] Equation of a Circle
  • [2] Center of a Circle
  • [3] Radius of a Circle

Understanding the Basics of Circles

Circles are a fundamental concept in geometry, and understanding their properties and equations is essential for solving problems in mathematics and science. In this article, we will answer some frequently asked questions about circles, covering topics such as the equation of a circle, the center of a circle, and the radius of a circle.

Q&A

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

A: The equation of a circle in standard form is given by (x-h)^2 + (y-k)^2 = r^2, where (h,k) represents the coordinates of the center of the circle, and r is the radius of the circle.

Q: How do I identify the center of a circle from its equation?

A: To identify the center of a circle from its equation, you need to compare the given equation with the standard form of the equation of a circle and identify the values of h and k.

Q: What is the radius of the circle represented by the equation (x+9)^2 + (y-6)^2 = 10^2?

A: The radius of the circle represented by the equation (x+9)^2 + (y-6)^2 = 10^2 is 10.

Q: How do I find the distance between the center and a point on the circle?

A: To find the distance between the center and a point on the circle, you can use the distance formula: d = sqrt((x2-x1)^2 + (y2-y1)^2), where (x1,y1) is the center of the circle and (x2,y2) is the point on the circle.

Q: What is the circumference of a circle?

A: The circumference of a circle is given by the formula C = 2πr, where r is the radius of the circle.

Q: How do I find the area of a circle?

A: To find the area of a circle, you can use the formula A = πr^2, where r is the radius of the circle.

Q: What is the difference between a circle and an ellipse?

A: A circle is a set of points that are equidistant from a central point, while an ellipse is a set of points that are equidistant from two central points.

Q: How do I graph a circle?

A: To graph a circle, you can use the equation of the circle and plot the points on a coordinate plane. You can also use a graphing calculator or software to graph the circle.

Real-World Applications of Circles

Circles have many real-world applications, including:

  • Geometry and Architecture: Circles are used in the design of buildings, bridges, and other structures.
  • Physics and Engineering: Circles are used to model the motion of objects and to design systems such as gears and pulleys.
  • Computer Science: Circles are used in computer graphics and game development to create 3D models and animations.
  • Navigation and Transportation: Circles are used in navigation systems such as GPS and in the design of roads and highways.

Conclusion

In conclusion, circles are a fundamental concept in geometry and have many real-world applications. Understanding the equation of a circle, the center of a circle, and the radius of a circle is essential for solving problems in mathematics and science. We hope that this article has provided you with a better understanding of circles and their properties.

Frequently Asked Questions

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

A: The equation of a circle in standard form is given by (x-h)^2 + (y-k)^2 = r^2, where (h,k) represents the coordinates of the center of the circle, and r is the radius of the circle.

Q: How do I identify the center of a circle from its equation?

A: To identify the center of a circle from its equation, you need to compare the given equation with the standard form of the equation of a circle and identify the values of h and k.

Q: What is the radius of the circle represented by the equation (x+9)^2 + (y-6)^2 = 10^2?

A: The radius of the circle represented by the equation (x+9)^2 + (y-6)^2 = 10^2 is 10.

References

Related Topics

  • [1] Equation of a Circle
  • [2] Center of a Circle
  • [3] Radius of a Circle
  • [4] Circumference of a Circle
  • [5] Area of a Circle