What Is The Radius Of A Circle Whose Equation Is X 2 + Y 2 + 8 X − 6 Y + 21 = 0 X^2 + Y^2 + 8x - 6y + 21 = 0 X 2 + Y 2 + 8 X − 6 Y + 21 = 0 ?A. 2 Units B. 3 Units C. 4 Units D. 5 Units

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Introduction

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 of the circle. In this article, we will discuss 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 standard form $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center of the circle and $r$ is the radius of the circle. To find the radius of a circle, we need to rewrite the given equation in the standard form.

Rewriting the Equation in Standard Form

To rewrite the equation in standard form, 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.

Completing the Square for the $x$ Term

The coefficient of the $x$ term is $8$. Half of this coefficient is $4$, and the square of this value is $16$. We can add and subtract $16$ to the equation to complete the square for the $x$ term.

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

Completing the Square for the $y$ Term

The coefficient of the $y$ term is $-6$. Half of this coefficient is $-3$, and the square of this value is $9$. We can add and subtract $9$ to the equation to complete the square for the $y$ term.

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

Simplifying the Equation

Now that we have completed the square for both the $x$ and $y$ terms, we can simplify the equation by combining like terms.

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

Simplifying Further

We can simplify the equation further by combining the constants.

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

Rearranging the Equation

We can rearrange the equation to put it in the standard form.

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

Finding the Radius of the Circle

Now that we have rewritten the equation in standard form, we can find the radius of the circle by taking the square root of the right-hand side of the equation.

r = \sqrt{4} = 2

Conclusion

In this article, we discussed how to find the radius of a circle whose equation is given as $x^2 + y^2 + 8x - 6y + 21 = 0$. We completed the square for both the $x$ and $y$ terms, simplified the equation, and rearranged it to put it in the standard form. Finally, we found the radius of the circle by taking the square root of the right-hand side of the equation. The radius of the circle is $2$ units.

Final Answer

The final answer is $\boxed{2}$ units.

Introduction

In our previous article, we discussed how to find the radius of a circle whose equation is given as $x^2 + y^2 + 8x - 6y + 21 = 0$. In this article, we will answer some frequently asked questions (FAQs) about the radius 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 circumference of the circle.

Q: How do I find the radius of a circle?

A: To find the radius of a circle, you need to rewrite the equation of the circle in standard form, which is $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center of the circle and $r$ is the radius of the circle. Then, you can take the square root of the right-hand side of the equation to find the radius.

Q: What is the formula for finding the radius of a circle?

A: The formula for finding the radius of a circle is $r = \sqrt{(x - h)^2 + (y - k)^2}$, where $(h, k)$ is the center of the circle.

Q: Can I find the radius of a circle if I don't have the equation of the circle?

A: No, you cannot find the radius of a circle if you don't have the equation of the circle. The equation of the circle is necessary to find the radius.

Q: How do I know if a circle is a perfect circle or not?

A: A circle is a perfect circle if its equation is in the standard form $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center of the circle and $r$ is the radius of the circle. If the equation is not in this form, then the circle is not a perfect circle.

Q: Can I find the radius of a circle if it is not a perfect circle?

A: Yes, you can find the radius of a circle even if it is not a perfect circle. However, the equation of the circle will not be in the standard form, and you will need to use other methods to find the radius.

Q: What is the difference between the radius and the diameter of a circle?

A: The radius of a circle is the distance from the center of the circle to any point on the circumference of the circle, while the diameter of a circle is twice the radius.

Q: Can I find the diameter of a circle if I know the radius?

A: Yes, you can find the diameter of a circle if you know the radius. The diameter is simply twice the radius.

Q: What is the relationship between the radius and the circumference of a circle?

A: The circumference of a circle is equal to $2\pi r$, where $r$ is the radius of the circle.

Q: Can I find the circumference of a circle if I know the radius?

A: Yes, you can find the circumference of a circle if you know the radius. The circumference is simply $2\pi r$.

Conclusion

In this article, we answered some frequently asked questions (FAQs) about the radius of a circle. We discussed how to find the radius of a circle, the formula for finding the radius, and the relationship between the radius and the diameter and circumference of a circle.

Final Answer

The final answer is that the radius of a circle is a fundamental concept in mathematics that is used to describe the size and shape of a circle. It can be found by rewriting the equation of the circle in standard form and taking the square root of the right-hand side of the equation.