The Equation Below Describes A Circle. What Are The Coordinates Of The Center Of The Circle?$\[(x-4)^2+(y+12)^2=17^2\\]A. \[$(4,-12)\$\]B. \[$(-4,-12)\$\]C. \[$(4,12)\$\]D. \[$(-4,12)\$\]
Introduction
In mathematics, a circle is a set of points that are all equidistant from a central point called the center. The equation of a circle is a fundamental concept in geometry and is used to describe the shape and size of a circle. In this article, we will explore the equation of a circle and how to find the coordinates of its center.
The Standard Equation of a Circle
The standard equation of a circle 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 represents the radius of the circle.
The Given Equation
The equation given in the problem is:
(x - 4)^2 + (y + 12)^2 = 17^2
Finding the Center of the Circle
To find the coordinates of the center of the circle, we need to identify the values of h and k in the equation. In this case, we can see that the equation is in the form (x - h)^2 + (y - k)^2 = r^2, where h = 4 and k = -12.
Why is this the case?
When we expand the equation (x - 4)^2 + (y + 12)^2 = 17^2, we get:
x^2 - 8x + 16 + y^2 + 24y + 144 = 289
Simplifying the equation, we get:
x^2 - 8x + y^2 + 24y = 29
Now, we can complete the square for the x and y terms:
(x - 4)^2 - 16 + (y + 12)^2 - 144 = 29
Simplifying further, we get:
(x - 4)^2 + (y + 12)^2 = 189
Comparing this with the standard equation of a circle, we can see that h = 4 and k = -12.
Conclusion
In conclusion, the coordinates of the center of the circle are (4, -12). This is because the equation of the circle is in the form (x - h)^2 + (y - k)^2 = r^2, where h = 4 and k = -12.
Answer
The correct answer is:
A. (4, -12)
Why is this the correct answer?
This is the correct answer because the equation of the circle is in the form (x - h)^2 + (y - k)^2 = r^2, where h = 4 and k = -12. This means that the coordinates of the center of the circle are (4, -12).
What is the significance of this result?
This result is significant because it shows how to find the coordinates of the center of a circle given its equation. This is an important concept in geometry and is used in a variety of applications, including physics, engineering, and computer science.
What are some common applications of this concept?
Some common applications of this concept include:
- Physics: The concept of a circle is used to describe the motion of objects in circular orbits.
- Engineering: The concept of a circle is used to design and build circular structures, such as bridges and tunnels.
- Computer Science: The concept of a circle is used in computer graphics and game development to create circular shapes and objects.
What are some common mistakes to avoid when working with circles?
Some common mistakes to avoid when working with circles include:
- Not completing the square: Failing to complete the square when working with the equation of a circle can lead to incorrect results.
- Not identifying the center: Failing to identify the center of the circle can lead to incorrect results.
- Not using the correct formula: Using the wrong formula for the equation of a circle can lead to incorrect results.
What are some common tips for working with circles?
Some common tips for working with circles include:
- Use the standard equation: Using the standard equation of a circle can make it easier to work with and understand.
- Complete the square: Completing the square when working with the equation of a circle can help to identify the center and radius of the circle.
- Use the correct formula: Using the correct formula for the equation of a circle can help to ensure accurate results.
Conclusion
Q: What is the equation of a circle?
A: The equation of a circle is a fundamental concept in geometry and is used to describe the shape and size of a circle. The standard equation of a circle 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 represents the radius of the circle.
Q: How do I find the center of a circle given its equation?
A: To find the center of a circle given its equation, you need to identify the values of h and k in the equation. In the standard equation of a circle, h and k are the values that are subtracted from x and y, respectively.
Q: What is the significance of the center of a circle?
A: The center of a circle is a critical point in geometry and is used to describe the shape and size of a circle. The center of a circle is the point from which all points on the circle are equidistant.
Q: How do I find the radius of a circle given its equation?
A: To find the radius of a circle given its equation, you need to take the square root of the value on the right-hand side of the equation. In the standard equation of a circle, r is the square root of the value on the right-hand side.
Q: What is the relationship between the center and radius of a circle?
A: The center and radius of a circle are related in that the center is the point from which all points on the circle are equidistant, and the radius is the distance from the center to any point on the circle.
Q: Can a circle have a negative radius?
A: No, a circle cannot have a negative radius. The radius of a circle is always a positive value.
Q: Can a circle have a zero radius?
A: No, a circle cannot have a zero radius. A circle with a zero radius would be a point, not a circle.
Q: What is the equation of a circle with a center at (3, 4) and a radius of 5?
A: The equation of a circle with a center at (3, 4) and a radius of 5 is:
(x - 3)^2 + (y - 4)^2 = 25
Q: What is the equation of a circle with a center at (-2, 1) and a radius of 3?
A: The equation of a circle with a center at (-2, 1) and a radius of 3 is:
(x + 2)^2 + (y - 1)^2 = 9
Q: How do I graph a circle given its equation?
A: To graph a circle given its equation, you need to plot the center of the circle and then draw a circle with a radius equal to the value on the right-hand side of the equation.
Q: What are some common applications of the equation of a circle?
A: Some common applications of the equation of a circle include:
- Physics: The equation of a circle is used to describe the motion of objects in circular orbits.
- Engineering: The equation of a circle is used to design and build circular structures, such as bridges and tunnels.
- Computer Science: The equation of a circle is used in computer graphics and game development to create circular shapes and objects.
Q: What are some common mistakes to avoid when working with circles?
A: Some common mistakes to avoid when working with circles include:
- Not completing the square: Failing to complete the square when working with the equation of a circle can lead to incorrect results.
- Not identifying the center: Failing to identify the center of the circle can lead to incorrect results.
- Not using the correct formula: Using the wrong formula for the equation of a circle can lead to incorrect results.
Q: What are some common tips for working with circles?
A: Some common tips for working with circles include:
- Use the standard equation: Using the standard equation of a circle can make it easier to work with and understand.
- Complete the square: Completing the square when working with the equation of a circle can help to identify the center and radius of the circle.
- Use the correct formula: Using the correct formula for the equation of a circle can help to ensure accurate results.