Finding the Center and Radius of the Circle

Oct 13, 2025 by windailys.com 55 views

Introduction

In mathematics, a circle is a set of points that are all equidistant from a central point called the center. The standard form equation of a circle is a mathematical representation of a circle in the Cartesian coordinate system. In this article, we will discuss how to write the standard form equation of a circle with a center at (-2, 4) and passing through (-5, 7).

What is the Standard Form Equation of a Circle?

The standard form 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 of the circle.

Finding the Center and Radius of the Circle

To find the center and radius of the circle, we need to use the given information that the circle passes through the point (-5, 7) and has a center at (-2, 4).

Step 1: Find the Distance Between the Center and the Point

The distance between the center (-2, 4) and the point (-5, 7) can be found using the distance formula:

d = √((x2 - x1)^2 + (y2 - y1)^2)

where (x1, y1) is the center (-2, 4) and (x2, y2) is the point (-5, 7).

d = √((-5 - (-2))^2 + (7 - 4)^2) d = √((-3)^2 + (3)^2) d = √(9 + 9) d = √18

Step 2: Find the Radius of the Circle

The radius of the circle is equal to the distance between the center and the point:

r = √18

Step 3: Write the Standard Form Equation of the Circle

Now that we have the center and radius of the circle, we can write the standard form equation of the circle:

(x - (-2))^2 + (y - 4)^2 = (√18)^2

(x + 2)^2 + (y - 4)^2 = 18

Conclusion

In this article, we discussed how to write the standard form equation of a circle with a center at (-2, 4) and passing through (-5, 7). We found the distance between the center and the point, the radius of the circle, and finally wrote the standard form equation of the circle. The standard form equation of the circle is:

(x + 2)^2 + (y - 4)^2 = 18

This equation represents a circle with a center at (-2, 4) and a radius of √18.

Example Problems

Problem 1

Write the standard form equation of a circle with a center at (3, 2) and passing through (6, 5).

Solution

To find the radius of the circle, we need to find the distance between the center (3, 2) and the point (6, 5):

d = √((6 - 3)^2 + (5 - 2)^2) d = √((3)^2 + (3)^2) d = √(9 + 9) d = √18

The radius of the circle is equal to the distance between the center and the point:

r = √18

Now that we have the center and radius of the circle, we can write the standard form equation of the circle:

(x - 3)^2 + (y - 2)^2 = (√18)^2

(x - 3)^2 + (y - 2)^2 = 18

Problem 2

Write the standard form equation of a circle with a center at (-1, 1) and passing through (0, 3).

Solution

To find the radius of the circle, we need to find the distance between the center (-1, 1) and the point (0, 3):

d = √((0 - (-1))^2 + (3 - 1)^2) d = √((1)^2 + (2)^2) d = √(1 + 4) d = √5

The radius of the circle is equal to the distance between the center and the point:

r = √5

Now that we have the center and radius of the circle, we can write the standard form equation of the circle:

(x - (-1))^2 + (y - 1)^2 = (√5)^2

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

A: The standard form equation of a circle is a mathematical representation of a circle in the Cartesian coordinate system. It 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 of the circle.

Q: How do I find the center and radius of a circle?

A: To find the center and radius of a circle, you need to use the given information that the circle passes through a point and has a center at a specific location. You can use the distance formula to find the distance between the center and the point, and then use this distance to find the radius of the circle.

Q: What is the distance formula?

A: The distance formula is given by:

d = √((x2 - x1)^2 + (y2 - y1)^2)

where (x1, y1) is the center of the circle and (x2, y2) is the point.

Q: How do I write the standard form equation of a circle?

A: To write the standard form equation of a circle, you need to substitute the values of the center and radius into the equation:

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

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

A: The standard form equation of a circle is important because it allows us to represent a circle in a mathematical way. It is used in various fields such as mathematics, physics, engineering, and computer science.

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

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

A = πr^2

where r is the radius of the circle.

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

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

C = 2πr

where r is the radius of the circle.

Q: What are some real-world applications of the standard form equation of a circle?

A: The standard form equation of a circle has many real-world applications, including:

Geometry: The standard form equation of a circle is used to represent a circle in a mathematical way.
Physics: The standard form equation of a circle is used to describe the motion of objects in a circular path.
Engineering: The standard form equation of a circle is used to design circular structures such as bridges, tunnels, and buildings.
Computer Science: The standard form equation of a circle is used in computer graphics to represent circular shapes.

Q: Can I use the standard form equation of a circle to solve problems involving circles?

A: Yes, you can use the standard form equation of a circle to solve problems involving circles. The standard form equation of a circle is a powerful tool that can be used to solve a wide range of problems involving circles.

Q: What are some common mistakes to avoid when using the standard form equation of a circle?

A: Some common mistakes to avoid when using the standard form equation of a circle include:

Incorrectly substituting values: Make sure to substitute the correct values into the equation.
Not simplifying the equation: Make sure to simplify the equation before solving it.
Not checking the solution: Make sure to check the solution to ensure that it is correct.

Conclusion

In this article, we have discussed the standard form equation of a circle and its applications. We have also answered some frequently asked questions about the standard form equation of a circle. We hope that this article has been helpful in understanding the standard form equation of a circle and its applications.