What Is The Radius Of A Circle Whose Equation Is $x^2 + Y^2 + 8x - 6y + 21 = 0$?A. 2 Units B. 3 Units C. 4 Units

What is the Radius of a Circle Whose Equation is $x^2 + y^2 + 8x - 6y + 21 = 0$?

In mathematics, the equation of a circle is a fundamental concept that is used to describe the shape and size 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. In this article, we will explore how to find the radius of a circle whose equation is given as $x^2 + y^2 + 8x - 6y + 21 = 0$.

Understanding 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 radius, we need to rewrite the given equation in the standard form of a circle.

Rewriting the Equation

To rewrite the equation, we need to complete the square for both the $x$ and $y$ terms. We can do this by adding and subtracting the square of half the coefficient of the $x$ term and the square of half the coefficient of the $y$ term.

x^2 + y^2 + 8x - 6y + 21 = 0

First, let's complete the square for the $x$ term. We can do this by adding and subtracting $(8/2)^2 = 16$.

x^2 + 8x + 16 - 16 + y^2 - 6y + 21 = 0

Next, let's complete the square for the $y$ term. We can do this by adding and subtracting $(-6/2)^2 = 9$.

x^2 + 8x + 16 - 16 + y^2 - 6y + 9 - 9 + 21 = 0

Now, we can rewrite the equation as:

(x + 4)^2 + (y - 3)^2 - 16 - 9 + 21 = 0

Simplifying the equation, we get:

(x + 4)^2 + (y - 3)^2 = 4

Finding the Radius

Now that we have rewritten the equation in the standard form of a circle, we can find the radius. The radius is the square root of the constant term on the right-hand side of the equation.

r = \sqrt{4} = 2

Therefore, the radius of the circle is 2 units.

In this article, we explored how to find the radius of a circle whose equation is given as $x^2 + y^2 + 8x - 6y + 21 = 0$. We rewrote the equation in the standard form of a circle and found the radius to be 2 units. This demonstrates the importance of understanding the equation of a circle and how to manipulate it to find the radius.

• Q: What is the equation of a circle? A: The 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 find the radius of a circle? A: To find the radius, you need to rewrite the equation in the standard form of a circle and then find the square root of the constant term on the right-hand side of the equation.
• Q: What is the radius of the circle in the given equation? A: The radius of the circle is 2 units.

• Equation of a circle
• Completing the square
• Radius of a circle

• [1] "Equation of a Circle" by Math Open Reference
• [2] "Completing the Square" by Khan Academy
• [3] "Radius of a Circle" by Wolfram MathWorld
  Q&A: Understanding the Equation of a Circle =============================================

In our previous article, we explored how to find the radius of a circle whose equation is given as $x^2 + y^2 + 8x - 6y + 21 = 0$. We rewrote the equation in the standard form of a circle and found the radius to be 2 units. In this article, we will answer some frequently asked questions about the equation of a circle and provide additional information to help you better understand this concept.

Q: What is the equation of a circle?

A: The 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 find the center of a circle?

A: To find the center of a circle, you need to rewrite the equation in the standard form of a circle. The center of the circle is given by the values of $h$ and $k$ 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 rewrite the equation in the standard form of a circle and then find 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 between the two equations is that the equation of a circle has a constant radius, while the equation of an ellipse has a variable radius.

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 the formula $A = \pi r^2$, where $r$ is the radius of the circle.

Q: How do I graph a circle?

A: To graph a circle, you need to find the center of the circle and the radius. You can then use a compass or a graphing calculator to draw 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 the formula $C = 2\pi r$, where $r$ is the radius of the circle.

Q: What is the relationship between the equation of a circle and the equation of a sphere?

A: The equation of a sphere is given by $(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2$, where $(h, k, l)$ is the center of the sphere and $r$ is the radius. The equation of a sphere is similar to the equation of a circle, but it includes an additional term for the $z$-coordinate.

In this article, we answered some frequently asked questions about the equation of a circle and provided additional information to help you better understand this concept. We hope that this article has been helpful in clarifying any confusion you may have had about the equation of a circle.

• Q: What is the equation of a circle? A: The 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 find the center of a circle? A: To find the center of a circle, you need to rewrite the equation in the standard form of a circle. The center of the circle is given by the values of $h$ and $k$ 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 rewrite the equation in the standard form of a circle and then find the square root of the constant term on the right-hand side of the equation.

• Equation of a circle
• Completing the square
• Radius of a circle
• Area of a circle
• Circumference of a circle
• Equation of an ellipse
• Equation of a sphere

• [1] "Equation of a Circle" by Math Open Reference
• [2] "Completing the Square" by Khan Academy
• [3] "Radius of a Circle" by Wolfram MathWorld
• [4] "Area of a Circle" by Math Is Fun
• [5] "Circumference of a Circle" by Math Is Fun