Select The Correct Answer.Circle { F $}$ Is Represented By The Equation { (x+6) 2+(y+8) 2=9$}$. What Is The Length Of The Radius Of Circle { F $}$?A. 3 B. 9 C. 10 D. 81

In mathematics, the equation of a circle is a fundamental concept that is used to represent the shape and size of a circle. The general equation of a circle is given by $(x-h{)}^2+(y-k{)}^2=r^2$(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 focus on understanding the equation of a circle and how to find the length of the radius.

The Equation of Circle { F $}$

The equation of circle { F $}$ is given by ${(x+6)^2+(y+8)^2=9}$. To find the length of the radius, we need to compare this equation with the general equation of a circle. By comparing the two equations, we can see that the center of the circle is ${(-6,-8)}$ and the radius is ${r}$.

Finding the Length of the Radius

To find the length of the radius, we need to find the value of ${r}$. We can do this by taking the square root of the right-hand side of the equation. The square root of 9 is 3, so the radius of the circle is ${r=3}$.

Conclusion

In this article, we have learned how to find the length of the radius of a circle using its equation. We have also seen how to compare the equation of a circle with the general equation of a circle to find the center and radius of the circle. The length of the radius of circle { F $}$ is 3.

Step-by-Step Solution

1. Step 1: Write down the equation of the circle.
2. Step 2: Compare the equation with the general equation of a circle to find the center and radius of the circle.
3. Step 3: Take the square root of the right-hand side of the equation to find the radius of the circle.

Common Mistakes

1. Mistake 1: Not comparing the equation of the circle with the general equation of a circle.
2. Mistake 2: Not taking the square root of the right-hand side of the equation to find the radius of the circle.

Real-World Applications

1. Application 1: Finding the length of the radius of a circle is important in architecture and engineering to design buildings and bridges.
2. Application 2: Finding the length of the radius of a circle is also important in physics to calculate the circumference and area of a circle.

Practice Problems

1. Problem 1: Find the length of the radius of the circle with the equation ${(x-4)^2+(y+2)^2=16}$.
2. Problem 2: Find the length of the radius of the circle with the equation ${(x+2)^2+(y-3)^2=25}$.

Answer Key

1. Answer 1: The length of the radius of the circle is 4.
2. Answer 2: The length of the radius of the circle is 5.

Conclusion

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Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about finding the length of the radius of a circle.

Q1: What is the general equation of a circle?

A1: 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.

Q2: How do I find the length of the radius of a circle?

A2: To find the length of the radius of a circle, you need to compare the equation of the circle with the general equation of a circle. Then, take the square root of the right-hand side of the equation to find the radius of the circle.

Q3: What is the center of the circle?

A3: The center of the circle is the point ${(h,k)}$ in the general equation of a circle.

Q4: How do I find the center of the circle?

A4: To find the center of the circle, you need to compare the equation of the circle with the general equation of a circle. The values of ${h}$ and ${k}$ in the general equation will give you the coordinates of the center of the circle.

Q5: What is the radius of the circle?

A5: The radius of the circle is the value of ${r}$ in the general equation of a circle.

Q6: How do I find the radius of the circle?

A6: To find the radius of the circle, you need to take the square root of the right-hand side of the equation of the circle.

Q7: What is the relationship between the radius and the diameter of a circle?

A7: The diameter of a circle is twice the radius of the circle.

Q8: How do I find the diameter of a circle?

A8: To find the diameter of a circle, you need to multiply the radius of the circle by 2.

Q9: What is the circumference of a circle?

A9: The circumference of a circle is given by the formula ${C=2\pi r}$, where ${r}$ is the radius of the circle.

Q10: How do I find the circumference of a circle?

A10: To find the circumference of a circle, you need to multiply the radius of the circle by 2 and then multiply the result by ${\pi}$.

Conclusion

In this article, we have answered some of the most frequently asked questions about finding the length of the radius of a circle. We have also provided information about the center and radius of a circle, as well as the relationship between the radius and the diameter of a circle. We hope that this article has been helpful in understanding the concept of finding the length of the radius of a circle.

Practice Problems

1. Problem 1: Find the length of the radius of the circle with the equation ${(x-3)^2+(y+1)^2=4}$.
2. Problem 2: Find the length of the radius of the circle with the equation ${(x+4)^2+(y-2)^2=9}$.

Answer Key

1. Answer 1: The length of the radius of the circle is 2.
2. Answer 2: The length of the radius of the circle is 3.

Real-World Applications

1. Application 1: Finding the length of the radius of a circle is important in architecture and engineering to design buildings and bridges.
2. Application 2: Finding the length of the radius of a circle is also important in physics to calculate the circumference and area of a circle.

Common Mistakes

1. Mistake 1: Not comparing the equation of the circle with the general equation of a circle.
2. Mistake 2: Not taking the square root of the right-hand side of the equation to find the radius of the circle.

Step-by-Step Solution

1. Step 1: Write down the equation of the circle.
2. Step 2: Compare the equation with the general equation of a circle to find the center and radius of the circle.
3. Step 3: Take the square root of the right-hand side of the equation to find the radius of the circle.