Which Equation Represents A Circle That Contains The Point { (-5,-3)$}$ And Has A Center At { (-2,1)$}$?Distance Formula: { \sqrt{\left(x_2-x_1\right) 2+\left(y_2-y_1\right) 2}$}$A. { (x-1) 2+(y+2) 2=25$}$B.

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 are given the center of the circle ${(-2,1)}$ and a point on the circle ${(-5,-3)}$. We need to find the equation of the circle that contains this point and has the given center.

Calculating the Radius

To find the equation of the circle, we need to calculate the radius, which is the distance between the center and the point on the circle. Using the distance formula, we can calculate the radius as follows:

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

Simplifying the expression, we get:

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

${\sqrt{9 + 16}}$

${\sqrt{25}}$

${5}$

Therefore, the radius of the circle is ${5}$ units.

Finding the Equation of the Circle

Now that we have the radius, we can find the equation of the circle using the standard form ${(x-h)^2 + (y-k)^2 = r^2}$. Plugging in the values, we get:

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

Simplifying the expression, we get:

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

Therefore, the equation of the circle that contains the point ${(-5,-3)}$ and has a center at ${(-2,1)}$ is ${(x+2)^2 + (y-1)^2 = 25}$.

Conclusion

In this article, we explored which equation represents a circle that contains the point ${(-5,-3)}$ and has a center at ${(-2,1)}$. We used the distance formula to calculate the radius of the circle and then found the equation of the circle using the standard form. The equation of the circle is ${(x+2)^2 + (y-1)^2 = 25}$.

Discussion

• What is the significance of the center of a circle in mathematics?
• How does the distance formula help in finding the equation of a circle?
• Can you think of any real-world applications of the equation of a circle?

Answer Key

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

Note

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Introduction

In our previous article, we explored which equation represents a circle that contains the point ${(-5,-3)}$ and has a center at ${(-2,1)}$. We used the distance formula to calculate the radius of the circle and then found the equation of the circle using the standard form. In this article, we will answer some frequently asked questions about circles and their equations.

Q: What is the significance of the center of a circle in mathematics?

A: The center of a circle is a crucial concept in mathematics. It is the point from which all points on the circle are equidistant. The center is also the point around which the circle is symmetric. In other words, if you draw a line from the center to any point on the circle, the line will be perpendicular to the circle at that point.

Q: How does the distance formula help in finding the equation of a circle?

A: The distance formula is a powerful tool in finding the equation of a circle. It helps us calculate the distance between two points in a coordinate plane. By using the distance formula, we can find the radius of the circle, which is the distance between the center and any point on the circle. With the radius and the center, we can then find the equation of the circle using the standard form.

Q: Can you think of any real-world applications of the equation of a circle?

A: Yes, there are many real-world applications of the equation of a circle. For example, in engineering, the equation of a circle is used to design circular structures such as bridges, tunnels, and pipes. In computer graphics, the equation of a circle is used to create circular shapes and patterns. In medicine, the equation of a circle is used to model the shape of organs and tissues.

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

A: The standard form of the equation of a circle is ${(x-h)^2 + (y-k)^2 = r^2}$, where ${(h,k)}$ is the center of the circle and ${r}$ is the radius.

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

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

Q: Can you give an example of how to find the equation of a circle using the standard form?

A: Let's say we have a circle with a center at ${(3,4)}$ and a point on the circle at ${(6,8)}$. To find the equation of the circle, we can use the distance formula to calculate the radius:

${\sqrt{(6-3)^2 + (8-4)^2}}$

Simplifying the expression, we get:

${\sqrt{3^2 + 4^2}}$

${\sqrt{9 + 16}}$

${\sqrt{25}}$

${5}$

Therefore, the radius of the circle is ${5}$ units. Now, we can use the standard form to find the equation of the circle:

${(x-3)^2 + (y-4)^2 = 5^2}$

Simplifying the expression, we get:

${(x-3)^2 + (y-4)^2 = 25}$

Therefore, the equation of the circle is ${(x-3)^2 + (y-4)^2 = 25}$.

Conclusion

In this article, we answered some frequently asked questions about circles and their equations. We discussed the significance of the center of a circle, how the distance formula helps in finding the equation of a circle, and real-world applications of the equation of a circle. We also provided an example of how to find the equation of a circle using the standard form.