Find The Equation Of A Circle, $\odot N$, That Passes Through The Point $(2, -2)$ And Has Its Center At $N(-1, 2)$.

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 is a mathematical representation of this concept, and it is used to describe the shape and size of a circle. In this article, we will discuss how to find the equation of a circle that passes through a given point and has its center at a specified location.

Understanding the Basics

Before we dive into the problem, let's review some basic concepts related to circles. 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.

The Problem

We are given that the circle, $\odot N$, passes through the point $(2, -2)$ and has its center at $N(-1, 2)$. Our goal is to find the equation of this circle.

Step 1: Identify the Center and the Point

The center of the circle is given as $N(-1, 2)$, and the point through which the circle passes is $(2, -2)$. We can use this information to find the radius of the circle.

Step 2: Find the Radius

To find the radius, we can use the distance formula between the center and the point:

$$r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

where $(x_1, y_1)$ is the center and $(x_2, y_2)$ is the point.

Plugging in the values, we get:

$$r = \sqrt{(-1 - 2)^2 + (2 - (-2))^2}$$

$$r = \sqrt{(-3)^2 + (4)^2}$$

$$r = \sqrt{9 + 16}$$

$$r = \sqrt{25}$$

$$r = 5$$

Step 3: Write the Equation of the Circle

Now that we have the center and the radius, we can write the equation of the circle using the general equation:

$$(x - h)^2 + (y - k)^2 = r^2$$

Plugging in the values, we get:

$$(x - (-1))^2 + (y - 2)^2 = 5^2$$

$$(x + 1)^2 + (y - 2)^2 = 25$$

Conclusion

In this article, we discussed how to find the equation of a circle that passes through a given point and has its center at a specified location. We used the distance formula to find the radius of the circle and then wrote the equation of the circle using the general equation. The final equation of the circle is:

$$(x + 1)^2 + (y - 2)^2 = 25$$

This equation represents the circle $\odot N$ that passes through the point $(2, -2)$ and has its center at $N(-1, 2)$.

Example Use Cases

The equation of a circle has many practical applications in mathematics and real-world problems. Here are a few examples:

• Geometry: The equation of a circle is used to describe the shape and size of a circle, which is a fundamental concept in geometry.
• Physics: The equation of a circle is used to describe the motion of objects in circular paths, such as the orbit of a planet around a star.
• Engineering: The equation of a circle is used to design and optimize circular structures, such as bridges and tunnels.

Tips and Variations

Here are a few tips and variations to keep in mind when working with the equation of a circle:

• Center-Radius Form: The equation of a circle can also be written in the center-radius form: $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center and $r$ is the radius.
• Standard Form: The equation of a circle can also be written in the standard form: $x^2 + y^2 + Dx + Ey + F = 0$, where $D$, $E$, and $F$ are constants.
• Parametric Form: The equation of a circle can also be written in the parametric form: $x = r \cos \theta$ and $y = r \sin \theta$, where $r$ is the radius and $\theta$ is the angle.

Conclusion

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 find the radius of a circle?

A: To find the radius of a circle, you can use the distance formula between the center and a point on the circle:

$$r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

where $(x_1, y_1)$ is the center and $(x_2, y_2)$ is the point.

Q: What is the center-radius form of the equation of a circle?

A: The center-radius form of the equation of a circle is:

$$(x - h)^2 + (y - k)^2 = r^2$$

where $(h, k)$ is the center and $r$ is the radius.

Q: What is the standard form of the equation of a circle?

A: The standard form of the equation of a circle is:

$$x^2 + y^2 + Dx + Ey + F = 0$$

where $D$, $E$, and $F$ are constants.

Q: How do I convert the standard form to the center-radius form?

A: To convert the standard form to the center-radius form, you can complete the square:

$$x^2 + y^2 + Dx + Ey + F = 0$$

$$(x + \frac{D}{2})^2 + (y + \frac{E}{2})^2 = \frac{D^2}{4} + \frac{E^2}{4} - F$$

Q: What is the parametric form of the equation of a circle?

A: The parametric form of the equation of a circle is:

$$x = r \cos \theta$$

$$y = r \sin \theta$$

where $r$ is the radius and $\theta$ is the angle.

Q: How do I find the equation of a circle that passes through two points?

A: To find the equation of a circle that passes through two points, you can use the following steps:

1. Find the midpoint of the two points.
2. Find the distance between the two points.
3. Use the distance formula to find the radius of the circle.
4. Write the equation of the circle using the center-radius form.

Q: How do I find the equation of a circle that passes through three points?

A: To find the equation of a circle that passes through three points, you can use the following steps:

1. Find the equation of the circle that passes through two of the points.
2. Use the equation of the circle to find the third point.
3. Use the third point to find the equation of the circle.

Q: What are some common applications of the equation of a circle?

A: The equation of a circle has many practical applications in mathematics and real-world problems, including:

• Geometry: The equation of a circle is used to describe the shape and size of a circle, which is a fundamental concept in geometry.
• Physics: The equation of a circle is used to describe the motion of objects in circular paths, such as the orbit of a planet around a star.
• Engineering: The equation of a circle is used to design and optimize circular structures, such as bridges and tunnels.

Conclusion

In conclusion, the equation of a circle is a fundamental concept in mathematics that has many practical applications in geometry, physics, and engineering. By understanding how to find the equation of a circle, we can describe the shape and size of a circle and solve problems involving circular structures.