The Center and Radius of the Circle

Oct 13, 2025 by magicalceylontours.com 24 views

Give the Center and Radius of the Circle Described by the Equation and Graph the Equation

In mathematics, a circle is a set of points that are equidistant from a central point called the center. The equation of a circle can be written in the form (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 give the center and radius of the circle described by the equation (x+5)^2+y2=4 and graph the equation. We will also use the graph to identify the domain and range of the circle.

The Equation of a Circle

The equation of a circle can be written in the form (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle and r is the radius. To find the center and radius of the circle, we need to rewrite the equation in this form.

Rewriting the Equation

Let's rewrite the equation (x+5)^2+y2=4 in the form (x - h)^2 + (y - k)^2 = r^2.

(x+5)^2+y2=4

(x+5)^2 = 4 - y^2

x^2 + 10x + 25 = 4 - y^2

x^2 + 10x + 21 = -y^2

x^2 + 10x + 21 + y^2 = 0

(x+5)^2 + y^2 = 21

Now we can see that the equation is in the form (x - h)^2 + (y - k)^2 = r^2, where h = -5 and k = 0.

The Center and Radius of the Circle

The center of the circle is (h, k) = (-5, 0) and the radius is r = √21.

Graphing the Equation

To graph the equation, we can use the following steps:

Plot the center of the circle, which is (-5, 0).
Draw a circle with the center at (-5, 0) and a radius of √21.

Here is a graph of the equation:

Graph of the Equation

The graph of the equation is a circle with the center at (-5, 0) and a radius of √21.

Domain and Range of the Circle

The domain of the circle is all real numbers x, and the range is all real numbers y such that y^2 ≤ 4.

In this article, we gave the center and radius of the circle described by the equation (x+5)^2+y2=4 and graphed the equation. We also used the graph to identify the domain and range of the circle. The center of the circle is (-5, 0) and the radius is √21. The domain of the circle is all real numbers x, and the range is all real numbers y such that y^2 ≤ 4.

The final answer is: $\boxed{(-5, 0), \sqrt{21}}$
Q&A: Circle Equation and Graphing

In our previous article, we discussed the equation of a circle and how to graph it. We also identified the domain and range of the circle. In this article, we will answer some frequently asked questions about circle equations and graphing.

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

A: The general form of a circle equation is (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 a circle equation in the general form?

A: To rewrite a circle equation in the general form, you need to complete the square for both the x and y terms. This involves adding and subtracting the same value to both sides of the equation.

Q: What is the center of the circle described by the equation (x+5)^2+y2=4?

A: The center of the circle is (-5, 0).

Q: What is the radius of the circle described by the equation (x+5)^2+y2=4?

A: The radius of the circle is √21.

Q: How do I graph a circle equation?

A: To graph a circle equation, you need to plot the center of the circle and then draw a circle with the center at that point and a radius equal to the radius of the circle.

Q: What is the domain of a circle?

A: The domain of a circle is all real numbers x.

Q: What is the range of a circle?

A: The range of a circle is all real numbers y such that y^2 ≤ r^2, where r is the radius of 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 number.

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 single point, not a circle.

Q: How do I find the equation of a circle given its center and radius?

A: To find the equation of a circle given its center and radius, you can use the general form of a circle equation: (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 answered some frequently asked questions about circle equations and graphing. We hope that this article has been helpful in clarifying any confusion you may have had about circle equations and graphing.

The final answer is: $\boxed{(-5, 0), \sqrt{21}}$