Rewriting the Equation of the Circle

Oct 13, 2025 by pinkandtrash.com 121 views

In mathematics, the equation of a circle is a fundamental concept that is used to describe the shape and position of a circle in a two-dimensional plane. 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. In this article, we will consider a specific equation of a circle, $x^2 + y^2 - 2x - 8 = 0$ , and determine which statements are true.

Rewriting the Equation of the Circle

To understand the equation of the circle, we need to rewrite it in the standard form. We can do this by completing the square for both the $x$ and $y$ terms.

x^2 + y^2 - 2x - 8 = 0

We can start by grouping the $x$ terms and the $y$ terms separately:

(x^2 - 2x) + (y^2) - 8 = 0

Next, we can complete the square for the $x$ terms by adding and subtracting $(2/2)^2 = 1$ :

(x^2 - 2x + 1) + (y^2) - 8 = 1

We can simplify the equation by combining the constants:

(x - 1)^2 + (y^2) - 9 = 0

Now, we can rewrite the equation in the standard form:

(x - 1)^2 + (y^2) = 9

Determining the Center and Radius of the Circle

From the rewritten equation, we can see that the center of the circle is at $(1, 0)$ and the radius is $\sqrt{9} = 3$ units.

Evaluating the Statements

Now that we have determined the center and radius of the circle, we can evaluate the statements:

A. The radius of the circle is 3 units.

This statement is true. We have determined that the radius of the circle is indeed 3 units.

B. The center of the circle lies on the $x$ -axis.

This statement is false. We have determined that the center of the circle is at $(1, 0)$ , which lies on the $y$ -axis, not the $x$ -axis.

C. The center of the circle lies on the $y$ -axis.

This statement is false. We have determined that the center of the circle is at $(1, 0)$ , which lies on the $x$ -axis, not the $y$ -axis.

Conclusion

In conclusion, we have determined that the radius of the circle is 3 units and the center of the circle lies on the $x$ -axis. We have also evaluated the statements and determined that only statement A is true.

Key Takeaways

The equation of a circle can be rewritten in the standard form by completing the square for both the $x$ and $y$ terms.
The center and radius of a circle can be determined from the rewritten equation.
The statements A, B, and C can be evaluated based on the center and radius of the circle.

References

[1] "Geometry: A Comprehensive Introduction" by Dan Pedoe
[2] "Mathematics for Computer Science" by Eric Lehman, F Thomson Leighton, and Albert R Meyer
Circle Equation Q&A =====================

In the previous article, we discussed the equation of a circle and determined the center and radius of a specific circle. In this article, we will answer some frequently asked questions about the equation of a circle.

Q: What is the general equation of a circle?

A: 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.

Q: How do I rewrite the equation of a circle in the standard form?

A: To rewrite the equation of a circle in the standard form, you need to complete the square for both the $x$ and $y$ terms. This involves adding and subtracting the square of half the coefficient of the $x$ or $y$ term.

Q: What is the center of a circle?

A: The center of a circle is the point in the middle of the circle, denoted by $(h, k)$ in the general equation of a 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, denoted by $r$ in the general equation of a 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, you need to rewrite the equation in the standard form by completing the square for both the $x$ and $y$ terms. The center of the circle will be the point $(h, k)$ and the radius will be the square root of the constant term on the right-hand side of the equation.

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

A: The equation of a circle is given by $(x - h)^2 + (y - k)^2 = r^2$ , while the equation of an ellipse is given by $\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$ . The main difference is that the equation of a circle has the same radius in all directions, while the equation of an ellipse has different radii in different directions.

Q: Can I use the equation of a circle to find the area of a circle?

A: Yes, you can use the equation of a circle to find the area of a circle. The area of a circle is given by $A = \pi r^2$ , where $r$ is the radius of the circle.

Q: Can I use the equation of a circle to find the circumference of a circle?

A: Yes, you can use the equation of a circle to find the circumference of a circle. The circumference of a circle is given by $C = 2\pi r$ , where $r$ is the radius of the circle.

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

A: The equation of a circle has many real-world applications, including:

Geometry and trigonometry: The equation of a circle is used to describe the shape and position of a circle in a two-dimensional plane.
Physics and engineering: The equation of a circle is used to describe the motion of objects in a circular path.
Computer graphics: The equation of a circle is used to create circular shapes and objects in computer graphics.
Mathematical modeling: The equation of a circle is used to model real-world phenomena, such as the motion of planets and the shape of a circle.

Conclusion

In conclusion, the equation of a circle is a fundamental concept in mathematics that has many real-world applications. We hope that this Q&A article has helped to clarify any questions you may have had about the equation of a circle.