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, a circle is a set of points that are all equidistant from a central point called the center. The distance from the center to any point on the circle is called the radius. In this article, we will learn how to identify the radius and the center of a circle given its equation.

The Equation 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.

Identifying the Center and Radius

Given the equation of a circle as $(x-5)^2 + y^2 = 81$, we can identify the center and radius by comparing it with the general equation of a circle.

• The term $(x-5)^2$ indicates that the center of the circle is at $(5, 0)$, since the general equation is $(x-h)^2$.
• The term $y^2$ indicates that the center of the circle is at $(0, 0)$, since the general equation is $(y-k)^2$.
• However, since the center is at $(5, 0)$, we can ignore the term $y^2$.
• The constant term on the right-hand side of the equation is $81$, which is equal to $r^2$. Therefore, the radius is $\sqrt{81} = 9$ units.

Conclusion

In conclusion, the radius of the circle is $\boxed{9}$ units, and the center of the circle is at $(\boxed{5}, \boxed{0})$.

Step-by-Step Solution

Here are the steps to identify the radius and the center of a circle given its equation:

1. Compare the given equation with the general equation of a circle.
2. Identify the center of the circle by comparing the terms $(x-h)^2$ and $(y-k)^2$ with the given equation.
3. Identify the radius by comparing the constant term on the right-hand side of the equation with $r^2$.

Example

Let's consider another example. Given the equation of a circle as $(x-2)^2 + (y+3)^2 = 16$, we can identify the center and radius as follows:

• The center of the circle is at $(2, -3)$, since the general equation is $(x-h)^2 + (y-k)^2$.
• The radius is $\sqrt{16} = 4$ units.

Tips and Tricks

Here are some tips and tricks to help you identify the radius and the center of a circle given its equation:

• Make sure to compare the given equation with the general equation of a circle.
• Identify the center of the circle by comparing the terms $(x-h)^2$ and $(y-k)^2$ with the given equation.
• Identify the radius by comparing the constant term on the right-hand side of the equation with $r^2$.

Common Mistakes

Here are some common mistakes to avoid when identifying the radius and the center of a circle given its equation:

• Not comparing the given equation with the general equation of a circle.
• Not identifying the center of the circle by comparing the terms $(x-h)^2$ and $(y-k)^2$ with the given equation.
• Not identifying the radius by comparing the constant term on the right-hand side of the equation with $r^2$.

Real-World Applications

The concept of identifying the radius and the center of a circle has many real-world applications, such as:

• Geometry and Trigonometry: The concept of identifying the radius and the center of a circle is used in geometry and trigonometry to solve problems involving circles and spheres.
• Physics and Engineering: The concept of identifying the radius and the center of a circle is used in physics and engineering to solve problems involving circular motion and rotation.
• Computer Science: The concept of identifying the radius and the center of a circle is used in computer science to solve problems involving graphics and game development.

Conclusion

Frequently Asked Questions

Here are some frequently asked questions about identifying the radius and the center 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 identify the center of a circle given its equation?

A: To identify the center of a circle given its equation, compare the terms $(x-h)^2$ and $(y-k)^2$ with the given equation. The values of $h$ and $k$ will give you the coordinates of the center of the circle.

Q: How do I identify the radius of a circle given its equation?

A: To identify the radius of a circle given its equation, compare the constant term on the right-hand side of the equation with $r^2$. The square root of this constant term will give you the radius of the circle.

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

A: The center of a circle is the point that is equidistant from all points on the circle, while the radius is the distance from the center to any point on the circle.

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

A: Yes, you can use the equation of a circle to find the area of the circle. The area of a circle is given by:

$$A = \pi r^2$$

where $r$ is the radius of the circle.

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

A: Yes, you can use the equation of a circle to find the circumference of the circle. The circumference of a circle is given by:

$$C = 2\pi r$$

where $r$ is the radius of the circle.

Q: What are some real-world applications of identifying the radius and the center of a circle?

A: Some real-world applications of identifying the radius and the center of a circle include:

• Geometry and Trigonometry: The concept of identifying the radius and the center of a circle is used in geometry and trigonometry to solve problems involving circles and spheres.
• Physics and Engineering: The concept of identifying the radius and the center of a circle is used in physics and engineering to solve problems involving circular motion and rotation.
• Computer Science: The concept of identifying the radius and the center of a circle is used in computer science to solve problems involving graphics and game development.

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

A: Yes, you can use the equation of a circle to find the equation of a tangent line to the circle. The equation of a tangent line to a circle is given by:

$$y - y_1 = m(x - x_1)$$

where $(x_1, y_1)$ is a point on the circle and $m$ is the slope of the tangent line.

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

A: Yes, you can use the equation of a circle to find the equation of a secant line to the circle. The equation of a secant line to a circle is given by:

$$y - y_1 = m(x - x_1)$$

where $(x_1, y_1)$ and $(x_2, y_2)$ are two points on the circle and $m$ is the slope of the secant line.

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

A: Yes, you can use the equation of a circle to find the equation of a chord of the circle. The equation of a chord of a circle is given by:

$$y - y_1 = m(x - x_1)$$

where $(x_1, y_1)$ and $(x_2, y_2)$ are two points on the chord and $m$ is the slope of the chord.

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

A: Yes, you can use the equation of a circle to find the equation of a diameter of the circle. The equation of a diameter of a circle is given by:

$$y - y_1 = m(x - x_1)$$

where $(x_1, y_1)$ and $(x_2, y_2)$ are two points on the diameter and $m$ is the slope of the diameter.

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

A: Yes, you can use the equation of a circle to find the equation of a radius of the circle. The equation of a radius of a circle is given by:

$$y - y_1 = m(x - x_1)$$

where $(x_1, y_1)$ is a point on the radius and $m$ is the slope of the radius.

Q: Can I use the equation of a circle to find the equation of a tangent line to the circle at a given point?

A: Yes, you can use the equation of a circle to find the equation of a tangent line to the circle at a given point. The equation of a tangent line to a circle at a given point is given by:

$$y - y_1 = m(x - x_1)$$

where $(x_1, y_1)$ is the given point and $m$ is the slope of the tangent line.

Q: Can I use the equation of a circle to find the equation of a secant line to the circle at two given points?

A: Yes, you can use the equation of a circle to find the equation of a secant line to the circle at two given points. The equation of a secant line to a circle at two given points is given by:

$$y - y_1 = m(x - x_1)$$

where $(x_1, y_1)$ and $(x_2, y_2)$ are the two given points and $m$ is the slope of the secant line.

Q: Can I use the equation of a circle to find the equation of a chord of the circle at two given points?

A: Yes, you can use the equation of a circle to find the equation of a chord of the circle at two given points. The equation of a chord of a circle at two given points is given by:

$$y - y_1 = m(x - x_1)$$

where $(x_1, y_1)$ and $(x_2, y_2)$ are the two given points and $m$ is the slope of the chord.

Q: Can I use the equation of a circle to find the equation of a diameter of the circle at two given points?

A: Yes, you can use the equation of a circle to find the equation of a diameter of the circle at two given points. The equation of a diameter of a circle at two given points is given by:

$$y - y_1 = m(x - x_1)$$

where $(x_1, y_1)$ and $(x_2, y_2)$ are the two given points and $m$ is the slope of the diameter.

Q: Can I use the equation of a circle to find the equation of a radius of the circle at a given point?

A: Yes, you can use the equation of a circle to find the equation of a radius of the circle at a given point. The equation of a radius of a circle at a given point is given by:

$$y - y_1 = m(x - x_1)$$

where $(x_1, y_1)$ is the given point and $m$ is the slope of the radius.

Q: Can I use the equation of a circle to find the equation of a tangent line to the circle at a given point and a given slope?

A: Yes, you can use the equation of a circle to find the equation of a tangent line to the circle at a given point and a given slope. The equation of a tangent line to a circle at a given point and a given slope is given by:

$$y - y_1 = m(x - x_1)$$

where $(x_1, y_1)$ is the given point and $m$ is the given slope.

Q: Can I use the equation of a circle to find the equation of a secant line to the circle at two given points and a given slope?

A: Yes, you can use the equation of a circle to find the equation of a secant line to the circle at two given points and a given slope. The equation of a secant line to a circle at two given points and a given slope is given by:

$$y - y_1 = m(x - x_1)$$

where $(x_1, y_1)$ and $(x_2, y_2)$ are the two given points and $m$ is the given slope.

Q: Can I use the equation of a circle to find the equation of a chord of the circle at two given points and a given slope?

A: Yes, you can use the equation of a circle to find the equation of a chord of the circle at two given points and a given slope. The equation of a chord of a circle at two given points and a given slope