Identify The Radius And The Center Of A Circle Whose Equation Is \[$(x-5)^2 + Y^2 = 81\$\].- The Radius Of The Circle Is \[$\square\$\] Units.- The Center Of The Circle Is At \[$(\square, \square)\$\].

Introduction

In mathematics, the equation of a circle is a fundamental concept that is used to describe the shape and position of a circle in a two-dimensional plane. The general equation of a circle is given by ${(x-h)^2 + (y-k)^2 = r^2}$, where ${(h, k)}$ represents the coordinates of the center of the circle, and ${r}$ represents the radius of the circle. In this article, we will focus on identifying the radius and center of a circle from its equation, using the given equation ${(x-5)^2 + y^2 = 81}$.

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 of the circle. The given equation ${(x-5)^2 + y^2 = 81}$ can be compared to the general equation of a circle to identify the values of ${h}$, ${k}$, and ${r}$. By comparing the two equations, we can see that ${h = 5}$, ${k = 0}$, and ${r^2 = 81}$.

Identifying the Radius of the Circle

To identify the radius of the circle, we need to find the value of ${r}$ from the equation ${r^2 = 81}$. Since ${r^2 = 81}$, we can take the square root of both sides to get ${r = \sqrt{81}}$. Simplifying the square root, we get ${r = 9}$. Therefore, the radius of the circle is ${9}$ units.

Identifying the Center of the Circle

To identify the center of the circle, we need to find the values of ${h}$ and ${k}$ from the equation ${(x-5)^2 + y^2 = 81}$. From the equation, we can see that ${h = 5}$ and ${k = 0}$. Therefore, the center of the circle is at ${(5, 0)}$.

Conclusion

In conclusion, we have identified the radius and center of a circle from its equation using the given equation ${(x-5)^2 + y^2 = 81}$. The radius of the circle is ${9}$ units, and the center of the circle is at ${(5, 0)}$. This demonstrates the importance of understanding the equation of a circle and how to identify its key components.

Example Problems

Problem 1

Find the radius and center of the circle whose equation is ${(x-2)^2 + (y-3)^2 = 16}$.

Solution

To find the radius and center of the circle, we need to compare the given equation to the general equation of a circle. By comparing the two equations, we can see that ${h = 2}$, ${k = 3}$, and ${r^2 = 16}$. Taking the square root of both sides, we get ${r = \sqrt{16}}$. Simplifying the square root, we get ${r = 4}$. Therefore, the radius of the circle is ${4}$ units. The center of the circle is at ${(2, 3)}$.

Problem 2

Find the radius and center of the circle whose equation is ${(x+1)^2 + y^2 = 25}$.

Solution

To find the radius and center of the circle, we need to compare the given equation to the general equation of a circle. By comparing the two equations, we can see that ${h = -1}$, ${k = 0}$, and ${r^2 = 25}$. Taking the square root of both sides, we get ${r = \sqrt{25}}$. Simplifying the square root, we get ${r = 5}$. Therefore, the radius of the circle is ${5}$ units. The center of the circle is at ${-1, 0}$.

Tips and Tricks

• To identify the radius and center of a circle from its equation, compare the given equation to the general equation of a circle.
• Identify the values of ${h}$, ${k}$, and ${r}$ from the equation.
• Take the square root of both sides to find the value of ${r}$.
• Simplify the square root to get the final value of ${r}$.

Conclusion

Frequently Asked Questions

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 ${(h, k)}$ represents the coordinates of the center of the circle, and ${r}$ represents the radius of the circle.

Q: How do I identify the radius and center of a circle from its equation?

A: To identify the radius and center of a circle from its equation, compare the given equation to the general equation of a circle. Identify the values of ${h}$, ${k}$, and ${r}$ from the equation. Take the square root of both sides to find the value of ${r}$. Simplify the square root to get the final value of ${r}$.

Q: What is the significance of the values of ${h}$ and ${k}$ in the equation of a circle?

A: The values of ${h}$ and ${k}$ represent the coordinates of the center of the circle. They are used to locate the center of the circle in the two-dimensional plane.

Q: How do I find the value of ${r}$ from the equation of a circle?

A: To find the value of ${r}$, take the square root of both sides of the equation. Simplify the square root to get the final value of ${r}$.

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

A: The radius of a circle is related to the equation of a circle by the equation ${r^2 = (x-h)^2 + (y-k)^2}$. This equation shows that the square of the radius is equal to the sum of the squares of the differences between the coordinates of a point on the circle and the coordinates of the center of the circle.

Q: Can I use the equation of a circle to find the distance between two points on the circle?

A: Yes, you can use the equation of a circle to find the distance between two points on the circle. By substituting the coordinates of the two points into the equation of the circle, you can find the distance between them.

Q: What is the significance of the equation of a circle in real-world applications?

A: The equation of a circle has many real-world applications, including geometry, trigonometry, and engineering. It is used to describe the shape and position of a circle in a two-dimensional plane, and it is used to solve problems and apply mathematical concepts to real-world situations.

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. By substituting the value of the radius into the formula for the area of a circle, you can find the area of the circle.

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

A: The equation of a circle is related to the concept of pi by the formula ${C = 2\pi r}$, where ${C}$ is the circumference of the circle and ${r}$ is the radius of the circle. This formula shows that the circumference of a circle is equal to twice the product of the radius and pi.

Q: Can I use the equation of a circle to find the volume of a sphere?

A: Yes, you can use the equation of a circle to find the volume of a sphere. By substituting the value of the radius into the formula for the volume of a sphere, you can find the volume of the sphere.

Q: What is the significance of the equation of a circle in the field of computer science?

A: The equation of a circle has many applications in the field of computer science, including computer graphics, game development, and computer vision. It is used to describe the shape and position of a circle in a two-dimensional plane, and it is used to solve problems and apply mathematical concepts to real-world situations.

Q: Can I use the equation of a circle to find the perimeter of a circle?

A: Yes, you can use the equation of a circle to find the perimeter of a circle. By substituting the value of the radius into the formula for the perimeter of a circle, you can find the perimeter of the circle.

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

A: The equation of a circle is related to the concept of trigonometry by the formulas for the sine, cosine, and tangent of an angle. These formulas show that the sine, cosine, and tangent of an angle are related to the coordinates of a point on the unit circle.

Q: Can I use the equation of a circle to find the surface area of a sphere?

A: Yes, you can use the equation of a circle to find the surface area of a sphere. By substituting the value of the radius into the formula for the surface area of a sphere, you can find the surface area of the sphere.

Q: What is the significance of the equation of a circle in the field of engineering?

A: The equation of a circle has many applications in the field of engineering, including mechanical engineering, electrical engineering, and civil engineering. It is used to describe the shape and position of a circle in a two-dimensional plane, and it is used to solve problems and apply mathematical concepts to real-world situations.

Q: Can I use the equation of a circle to find the diameter of a circle?

A: Yes, you can use the equation of a circle to find the diameter of a circle. By substituting the value of the radius into the formula for the diameter of a circle, you can find the diameter of the circle.

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

A: The equation of a circle is related to the concept of geometry by the formulas for the area and perimeter of a circle. These formulas show that the area and perimeter of a circle are related to the radius of the circle.

Q: Can I use the equation of a circle to find the volume of a cylinder?

A: Yes, you can use the equation of a circle to find the volume of a cylinder. By substituting the value of the radius into the formula for the volume of a cylinder, you can find the volume of the cylinder.

Q: What is the significance of the equation of a circle in the field of physics?

A: The equation of a circle has many applications in the field of physics, including classical mechanics, electromagnetism, and quantum mechanics. It is used to describe the shape and position of a circle in a two-dimensional plane, and it is used to solve problems and apply mathematical concepts to real-world situations.

Q: Can I use the equation of a circle to find the surface area of a cylinder?

A: Yes, you can use the equation of a circle to find the surface area of a cylinder. By substituting the value of the radius into the formula for the surface area of a cylinder, you can find the surface area of the cylinder.

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

A: The equation of a circle is related to the concept of calculus by the formulas for the area and perimeter of a circle. These formulas show that the area and perimeter of a circle are related to the radius of the circle.

Q: Can I use the equation of a circle to find the volume of a cone?

A: Yes, you can use the equation of a circle to find the volume of a cone. By substituting the value of the radius into the formula for the volume of a cone, you can find the volume of the cone.

Q: What is the significance of the equation of a circle in the field of computer-aided design (CAD)?

A: The equation of a circle has many applications in the field of computer-aided design (CAD), including computer-aided drafting and computer-aided manufacturing. It is used to describe the shape and position of a circle in a two-dimensional plane, and it is used to solve problems and apply mathematical concepts to real-world situations.

Q: Can I use the equation of a circle to find the surface area of a cone?

A: Yes, you can use the equation of a circle to find the surface area of a cone. By substituting the value of the radius into the formula for the surface area of a cone, you can find the surface area of the cone.

Q: What is the relationship between the equation of a circle and the concept of differential equations?

A: The equation of a circle is related to the concept of differential equations by the formulas for the area and perimeter of a circle. These formulas show that the area and perimeter of a circle are related to the radius of the circle.

Q: Can I use the equation of a circle to find the volume of a sphere?

A: Yes, you can use the equation of a circle to find the volume of a sphere. By substituting the value of the radius into the formula for the volume of a sphere, you can find the volume of the sphere.

Q: What is the significance of the equation of a circle in the field of data analysis?

A: The equation of a circle has many applications in the field of data analysis, including data visualization and data mining. It is used to describe the shape and position of a circle in a two-dimensional plane, and it is used to solve problems and apply mathematical concepts to real-world situations.

Q: Can I use the equation of a circle to find the surface area of a sphere?

A: Yes,