Using The Equation X 2 + Y 2 − 8 X + 2 Y = 8 X^2 + Y^2 - 8x + 2y = 8 X 2 + Y 2 − 8 X + 2 Y = 8 , Calculate The Center And Radius Of The Circle By Completing The Square.A. Center = (4, -1), Radius = 5 B. Center = (8, -2), Radius = 25 C. Center = (2, -8), Radius = 25 D. Center = (-4, 1),
Introduction
In mathematics, a circle is a set of points that are equidistant from a central point called the center. The distance from the center to any point on the circle is called the radius. In this article, we will use the equation of a circle to find its center and radius by completing the square.
The Equation of a Circle
The general equation of a circle is given by:
(x - h)^2 + (y - k)^2 = r^2
where (h, k) is the center of the circle and r is the radius.
Completing the Square
To find the center and radius of the circle, we need to complete the square. The equation of the circle is given by:
x^2 + y^2 - 8x + 2y = 8
We can start by grouping the x-terms and y-terms:
(x^2 - 8x) + (y^2 + 2y) = 8
Now, we need to add and subtract the square of half the coefficient of x and y to complete the square:
(x^2 - 8x + 16) + (y^2 + 2y + 1) = 8 + 16 + 1
This can be rewritten as:
(x - 4)^2 + (y + 1)^2 = 25
Finding the Center and Radius
Now that we have completed the square, we can see that the equation of the circle is in the form:
(x - h)^2 + (y - k)^2 = r^2
Comparing this with our equation, we can see that:
h = 4 k = -1 r^2 = 25
Therefore, the center of the circle is (4, -1) and the radius is √25 = 5.
Conclusion
In this article, we used the equation of a circle to find its center and radius by completing the square. We started with the general equation of a circle and completed the square to get the equation in the form (x - h)^2 + (y - k)^2 = r^2. We then compared this with our equation to find the center and radius of the circle. The center of the circle is (4, -1) and the radius is 5.
Answer
The correct answer is:
A. Center = (4, -1), Radius = 5
Discussion
This problem is a great example of how completing the square can be used to find the center and radius of a circle. By completing the square, we were able to rewrite the equation of the circle in the form (x - h)^2 + (y - k)^2 = r^2, which made it easy to find the center and radius. This technique is useful in many areas of mathematics, including algebra and geometry.
Related Topics
- Completing the square: This is a technique used to rewrite a quadratic equation in the form (x - h)^2 + (y - k)^2 = r^2.
- Circle equations: This is a topic that deals with the equations of circles and how to find their centers and radii.
- Algebra: This is a branch of mathematics that deals with the study of variables and their relationships.
References
- [1] "Completing the Square" by Math Open Reference
- [2] "Circle Equations" by Khan Academy
- [3] "Algebra" by Wikipedia
Completing the Square to Find the Center and Radius of a Circle: Q&A ====================================================================
Introduction
In our previous article, we used the equation of a circle to find its center and radius by completing the square. In this article, we will answer some frequently asked questions about completing the square and finding the center and radius of a circle.
Q: What is completing the square?
A: Completing the square is a technique used to rewrite a quadratic equation in the form (x - h)^2 + (y - k)^2 = r^2. This is useful in many areas of mathematics, including algebra and geometry.
Q: How do I complete the square?
A: To complete the square, you need to follow these steps:
- Group the x-terms and y-terms.
- Add and subtract the square of half the coefficient of x and y.
- Rewrite the equation in the form (x - h)^2 + (y - k)^2 = r^2.
Q: What is the center of a circle?
A: The center of a circle is the point that is equidistant from all points on the circle. It is represented by the coordinates (h, k) in the equation (x - h)^2 + (y - k)^2 = r^2.
Q: How do I find the center of a circle?
A: To find the center of a circle, you need to complete the square and rewrite the equation in the form (x - h)^2 + (y - k)^2 = r^2. The coordinates (h, k) will be the center of the circle.
Q: What is the radius of a circle?
A: The radius of a circle is the distance from the center of the circle to any point on the circle. It is represented by the value r in the equation (x - h)^2 + (y - k)^2 = r^2.
Q: How do I find the radius of a circle?
A: To find the radius of a circle, you need to complete the square and rewrite the equation in the form (x - h)^2 + (y - k)^2 = r^2. The value of r will be the radius of the circle.
Q: Can I use completing the square to find the center and radius of any circle?
A: Yes, you can use completing the square to find the center and radius of any circle. However, you need to make sure that the equation of the circle is in the form (x - h)^2 + (y - k)^2 = r^2.
Q: What are some common mistakes to avoid when completing the square?
A: Some common mistakes to avoid when completing the square include:
- Not grouping the x-terms and y-terms correctly.
- Not adding and subtracting the square of half the coefficient of x and y correctly.
- Not rewriting the equation in the form (x - h)^2 + (y - k)^2 = r^2.
Conclusion
In this article, we answered some frequently asked questions about completing the square and finding the center and radius of a circle. We hope that this article has been helpful in understanding the concept of completing the square and how to use it to find the center and radius of a circle.
Related Topics
- Completing the square: This is a technique used to rewrite a quadratic equation in the form (x - h)^2 + (y - k)^2 = r^2.
- Circle equations: This is a topic that deals with the equations of circles and how to find their centers and radii.
- Algebra: This is a branch of mathematics that deals with the study of variables and their relationships.
References
- [1] "Completing the Square" by Math Open Reference
- [2] "Circle Equations" by Khan Academy
- [3] "Algebra" by Wikipedia