Finding the Equation of the Circle

Oct 13, 2025 by leolavanderia.com.br 134 views

Introduction

In mathematics, a circle is a set of points that are all equidistant from a central point called the center. The equation of a circle can be represented in various forms, including the standard form, which 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. In this article, we will explore which equation represents a circle that contains the point ${(-5,-3)}$ and has a center at ${(-2,1)}$ .

Understanding the Problem

To find the equation of the circle, we need to use the distance formula, which is given by ${\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}}$ . This formula calculates the distance between two points ${(x_1, y_1)}$ and ${(x_2, y_2)}$ in a coordinate plane. In this case, we want to find the distance between the center of the circle ${(-2,1)}$ and the point ${(-5,-3)}$ .

Calculating the Distance

Using the distance formula, we can calculate the distance between the center of the circle and the point as follows:

${\sqrt{((-5)-(-2))^2 + ((-3)-(1))^2}}$

Simplifying the expression, we get:

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

Expanding the squares, we get:

${\sqrt{9 + 16}}$

Adding the numbers, we get:

${\sqrt{25}}$

Simplifying the square root, we get:

${5}$

Finding the Equation of the Circle

Now that we have the distance between the center of the circle and the point, we can use it to find the equation of the circle. The 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. In this case, the center of the circle is ${(-2,1)}$ and the radius is ${5}$ .

Substituting the values into the equation, we get:

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

Simplifying the equation, we get:

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

Comparing the Options

Now that we have the equation of the circle, we can compare it with the options given. The options are:

A. ${(x-1)^2 + (y+2)^2 = 25}$

B. ${(x+2)^2 + (y-1)^2 = 25}$

Comparing the options, we can see that option B matches the equation we derived.

Conclusion

In conclusion, the equation that represents a circle that contains the point ${(-5,-3)}$ and has a center at ${(-2,1)}$ is ${(x+2)^2 + (y-1)^2 = 25}$ . This equation is derived using the distance formula and the standard form of the equation of a circle.

Final Answer

The final answer is:

$${(x+2)^2 + (y-1)^2 = 25}$<br/> **Q&A: Understanding Circles and Their Equations**$

Introduction

In our previous article, we explored the equation of a circle that contains the point ${(-5,-3)}$ and has a center at ${(-2,1)}$ . In this article, we will answer some frequently asked questions about circles and their equations.

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

A: The standard form of the 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 equation of a circle if I know the center and a point on the circle?

A: To find the equation of a circle, you can use the distance formula to calculate the distance between the center and the point. Then, you can use the standard form of the equation of a circle and substitute the values of the center and the radius.

Q: What is the radius of a circle?

A: The radius of a circle is the distance from the center of the circle to any point on the circle. It is denoted by the letter ${r}$ .

Q: How do I find the radius of a circle if I know the center and a point on the circle?

A: To find the radius of a circle, you can use the distance formula to calculate the distance between the center and the point. This distance is equal to the radius of the circle.

Q: What is the center of a circle?

A: The center of a circle is the point from which all points on the circle are equidistant. It is denoted by the coordinates ${(h,k)}$ in the standard form of the equation of a circle.

Q: How do I find the center of a circle if I know the equation of the circle?

A: To find the center of a circle, you can look at the standard form of the equation of the circle and identify the values of ${h}$ and ${k}$ , which represent the coordinates of the center.

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

A: The center and the radius of a circle are related in that the radius is the distance from the center to any point on the circle. This means that if you know the center and the radius, you can find the equation of the circle.

Q: Can a circle have a negative radius?

A: No, a circle cannot have a negative radius. The radius of a circle is always a positive value, representing the distance from the center to any point on the circle.

Q: Can a circle have a zero radius?

A: No, a circle cannot have a zero radius. A circle with a zero radius would be a point, not a circle.

Conclusion

In conclusion, understanding the equation of a circle is essential in mathematics and other fields. By knowing the standard form of the equation of a circle and how to find the center and radius, you can solve problems involving circles and their equations.

Final Answer

The final answer is:

The standard form of the equation of a circle is ${(x-h)^2 + (y-k)^2 = r^2}$ .
To find the equation of a circle, use the distance formula to calculate the distance between the center and a point on the circle.
The radius of a circle is the distance from the center to any point on the circle.
The center of a circle is the point from which all points on the circle are equidistant.
The relationship between the center and the radius of a circle is that the radius is the distance from the center to any point on the circle.