A Circle Is Represented By The Equation X 2 + Y 2 − 16 X − 6 Y = − 24 X^2 + Y^2 - 16x - 6y = -24 X 2 + Y 2 − 16 X − 6 Y = − 24 . Complete The Square To Discover The Center And Radius Of The Circle.A. Center: ( − 16 , − 6 (-16, -6 ( − 16 , − 6 ], Radius: 8 8 8 B. Center: ( 8 , 3 (8, 3 ( 8 , 3 ], Radius: 7 7 7

Introduction

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 explore how to complete the square to find the center and radius of a circle given by the equation $x^2 + y^2 - 16x - 6y = -24$.

Completing the Square

To complete the square, we need to rewrite the equation in a way that allows us to easily identify the center and radius of the circle. We can start by grouping the $x$ and $y$ terms together:

$$x^2 - 16x + y^2 - 6y = -24$$

Next, we need to add and subtract the square of half the coefficient of each term to create a perfect square trinomial. For the $x$ term, we need to add and subtract $(16/2)^2 = 64$, and for the $y$ term, we need to add and subtract $(6/2)^2 = 9$:

$$x^2 - 16x + 64 - 64 + y^2 - 6y + 9 - 9 = -24$$

Now, we can rewrite the equation as:

$$(x^2 - 16x + 64) + (y^2 - 6y + 9) - 64 - 9 = -24$$

This can be simplified to:

$$(x - 8)^2 + (y - 3)^2 - 73 = -24$$

Finding the Center and Radius

Now that we have completed the square, we can easily identify the center and radius of the circle. The center of the circle is given by the values of $h$ and $k$ in the equation $(x - h)^2 + (y - k)^2 = r^2$. In this case, we have:

$$(x - 8)^2 + (y - 3)^2 = 73 + 24$$

This simplifies to:

$$(x - 8)^2 + (y - 3)^2 = 97$$

Therefore, the center of the circle is $(8, 3)$, and the radius is $\sqrt{97}$.

Conclusion

In this article, we have shown how to complete the square to find the center and radius of a circle given by the equation $x^2 + y^2 - 16x - 6y = -24$. We have seen that the center of the circle is $(8, 3)$, and the radius is $\sqrt{97}$. This demonstrates the power of completing the square in solving problems involving circles.

Discussion

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 case, we have:

$$(x - 8)^2 + (y - 3)^2 = 97$$

This means that the center of the circle is $(8, 3)$, and the radius is $\sqrt{97}$.

Why is Completing the Square Important?

Completing the square is an important technique in mathematics that allows us to rewrite an equation in a way that makes it easier to solve. It is particularly useful when working with quadratic equations, such as the equation of a circle. By completing the square, we can easily identify the center and radius of the circle, which is essential in many mathematical and real-world applications.

Real-World Applications

Completing the square has many real-world applications, including:

• Geometry: Completing the square is used to find the center and radius of a circle, which is essential in geometry.
• Physics: Completing the square is used to solve problems involving motion and energy.
• Engineering: Completing the square is used to design and optimize systems, such as electrical circuits and mechanical systems.

Conclusion

In conclusion, completing the square is an important technique in mathematics that allows us to rewrite an equation in a way that makes it easier to solve. It is particularly useful when working with quadratic equations, such as the equation of a circle. By completing the square, we can easily identify the center and radius of the circle, which is essential in many mathematical and real-world applications.

Final Answer

The final answer is:

• Center: $(8, 3)$
• Radius: $\sqrt{97}$
  Completing the Square: A Q&A Guide =====================================

Introduction

Completing the square is a powerful technique in mathematics that allows us to rewrite an equation in a way that makes it easier to solve. In our previous article, we showed how to complete the square to find the center and radius of a circle given by the equation $x^2 + y^2 - 16x - 6y = -24$. In this article, we will answer some frequently asked questions about completing the square.

Q: What is completing the square?

A: Completing the square is a technique used to rewrite a quadratic equation 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.

Q: Why is completing the square important?

A: Completing the square is important because it allows us to easily identify the center and radius of a circle, which is essential in many mathematical and real-world applications.

Q: How do I complete the square?

A: To complete the square, you need to follow these steps:

1. Group the $x$ and $y$ terms together.
2. Add and subtract the square of half the coefficient of each term to create a perfect square trinomial.
3. Simplify the equation to get the desired form.

Q: What are some common mistakes to avoid when completing the square?

A: Some common mistakes to avoid when completing the square include:

• Not adding and subtracting the square of half the coefficient of each term.
• Not simplifying the equation correctly.
• Not checking the equation for errors.

Q: Can I use completing the square to solve other types of equations?

A: Yes, you can use completing the square to solve other types of equations, such as quadratic equations and systems of equations.

Q: How do I know when to use completing the square?

A: You should use completing the square when you have a quadratic equation that can be rewritten in the form $(x - h)^2 + (y - k)^2 = r^2$. This is often the case when working with circles and other conic sections.

Q: Can I use a calculator to complete the square?

A: Yes, you can use a calculator to complete the square. However, it's often more helpful to do it by hand, as this will help you understand the underlying math and avoid errors.

Q: What are some real-world applications of completing the square?

A: Some real-world applications of completing the square include:

• Geometry: Completing the square is used to find the center and radius of a circle, which is essential in geometry.
• Physics: Completing the square is used to solve problems involving motion and energy.
• Engineering: Completing the square is used to design and optimize systems, such as electrical circuits and mechanical systems.

Conclusion

In conclusion, completing the square is a powerful technique in mathematics that allows us to rewrite an equation in a way that makes it easier to solve. By understanding how to complete the square, you can solve a wide range of mathematical problems and apply the technique to real-world applications.

Final Answer

The final answer is:

• Completing the square is a technique used to rewrite a quadratic equation in the form $(x - h)^2 + (y - k)^2 = r^2$.
• Completing the square is important because it allows us to easily identify the center and radius of a circle.
• To complete the square, you need to follow these steps: group the $x$ and $y$ terms together, add and subtract the square of half the coefficient of each term, and simplify the equation.