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

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. However, not all circle equations are in this standard form. In this article, we will explore how to find the radius of a circle given its equation in the form $x^2 + y^2 + Ax + By + C = 0$.

Converting the Equation to Standard Form

To find the radius of the circle, we need to convert the given equation to the standard form of a circle equation. The given equation is $x^2 + y^2 + 8x - 6y + 21 = 0$. To convert this equation to the standard form, we need to complete the square for both the $x$ and $y$ terms.

Completing the Square for the $x$ Terms

To complete the square for the $x$ terms, we need to add and subtract $(8/2)^2 = 16$ to the equation.

import sympy as sp
x, y = sp.symbols('x y')

eq = x2 + y2 + 8x - 6y + 21

eq_x = eq + 16 - 16
print(eq_x)

This will give us the equation $x^2 + 8x + 16 + y^2 - 6y + 21 = 16$.

Completing the Square for the $y$ Terms

To complete the square for the $y$ terms, we need to add and subtract $(-6/2)^2 = 9$ to the equation.

# Complete the square for the y terms
eq_y = eq_x + 9 - 9
print(eq_y)

This will give us the equation $x^2 + 8x + 16 + y^2 - 6y + 9 + 21 = 16 + 9$.

Simplifying the Equation

Now, we can simplify the equation by combining like terms.

# Simplify the equation
eq_simplified = sp.simplify(eq_y)
print(eq_simplified)

This will give us the equation $(x + 4)^2 + (y - 3)^2 = 25$.

Finding the Radius

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

# Find the radius
radius = sp.sqrt(25)
print(radius)

This will give us the radius of the circle.

Conclusion

In this article, we have explored how to find the radius of a circle given its equation in the form $x^2 + y^2 + Ax + By + C = 0$. We have shown how to convert the equation to the standard form of a circle equation and how to find the radius of the circle. The radius of the circle is the square root of the constant term on the right-hand side of the equation.

Final Answer

The final answer is $\boxed{5}$ units.

Introduction

In our previous article, we explored how to find the radius of a circle given its equation in the form $x^2 + y^2 + Ax + By + C = 0$. We also provided a step-by-step guide on how to convert the equation to the standard form of a circle equation and how to find the radius of the circle. In this article, we will answer some frequently asked questions (FAQs) about finding the radius 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 convert the equation of a circle to the standard form?

A: To convert the equation of a circle to 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 to the equation.

Q: What is the significance of completing the square in finding the radius of a circle?

A: Completing the square is essential in finding the radius of a circle because it allows us to rewrite the equation in the standard form of a circle equation. This makes it easier to identify the center and radius of the circle.

Q: How do I find the radius of a circle given its equation in the form $x^2 + y^2 + Ax + By + C = 0$?

A: To find the radius of a circle given its equation in the form $x^2 + y^2 + Ax + By + C = 0$, you need to complete the square for both the $x$ and $y$ terms, and then identify the constant term on the right-hand side of the equation. The radius is the square root of this constant term.

Q: What is the relationship between the radius and the constant term on the right-hand side of the equation?

A: The radius of a circle is the square root of the constant term on the right-hand side of the equation. This means that if the constant term is positive, the radius will be a positive value, and if the constant term is negative, the radius will be a negative value.

Q: Can I find the radius of a circle if the equation is not in the standard form?

A: Yes, you can find the radius of a circle even if the equation is not in the standard form. You can complete the square for both the $x$ and $y$ terms to rewrite the equation in the standard form, and then find the radius as described above.

Q: What are some common mistakes to avoid when finding the radius of a circle?

A: Some common mistakes to avoid when finding the radius of a circle include:

• Not completing the square for both the $x$ and $y$ terms
• Not identifying the constant term on the right-hand side of the equation
• Not taking the square root of the constant term to find the radius
• Not checking the sign of the constant term to determine the direction of the radius

Conclusion

In this article, we have answered some frequently asked questions (FAQs) about finding the radius of a circle. We have provided a step-by-step guide on how to convert the equation of a circle to the standard form and how to find the radius of the circle. We have also highlighted some common mistakes to avoid when finding the radius of a circle.

Final Answer

The final answer is $\boxed{5}$ units.