Finding the Equation of a Circle with a Given Radius and Center

Oct 13, 2025 by windailys.com 82 views

Finding the Equation of a Circle with a Given Radius and Center

In this article, we will explore the process of finding the equation of a circle with a given radius and center. The equation of a circle is a fundamental concept in mathematics, and it has numerous applications in various fields, including physics, engineering, and computer science. In this discussion, we will focus on finding the equation of a circle with a radius of 5, whose center lies on the y-axis and passes through the point (3, 2).

To find the equation of the circle, we need to understand the properties of a circle and the given information. The equation of a circle with center (h, k) and radius r is given by:

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

In this case, the radius of the circle is 5, and the center lies on the y-axis. This means that the x-coordinate of the center is 0, and the y-coordinate is unknown. The circle passes through the point (3, 2), which means that the coordinates of this point satisfy the equation of the circle.

Since the center of the circle lies on the y-axis, the x-coordinate of the center is 0. Let's denote the y-coordinate of the center as k. The equation of the circle can be written as:

(x - 0)^2 + (y - k)^2 = 5^2

Simplifying the equation, we get:

x^2 + (y - k)^2 = 25

Since the circle passes through the point (3, 2), we can substitute these coordinates into the equation:

(3)^2 + (2 - k)^2 = 25

Expanding the equation, we get:

9 + (2 - k)^2 = 25

Simplifying the equation, we get:

(2 - k)^2 = 16

Taking the square root of both sides, we get:

2 - k = ±4

Solving for k, we get:

k = 2 ± 4

k = -2 or k = 6

Since the center of the circle lies on the y-axis, the x-coordinate of the center is 0. Therefore, the center of the circle is (0, -2) or (0, 6).

Now that we have found the center of the circle, we can find the equation of the circle. Let's consider the two possible centers: (0, -2) and (0, 6).

Case 1: Center (0, -2)

The equation of the circle with center (0, -2) and radius 5 is:

(x - 0)^2 + (y - (-2))^2 = 5^2

Simplifying the equation, we get:

x^2 + (y + 2)^2 = 25

Case 2: Center (0, 6)

The equation of the circle with center (0, 6) and radius 5 is:

(x - 0)^2 + (y - 6)^2 = 5^2

Simplifying the equation, we get:

x^2 + (y - 6)^2 = 25

In this article, we have found the equation of a circle with a radius of 5, whose center lies on the y-axis and passes through the point (3, 2). We have considered two possible centers: (0, -2) and (0, 6), and found the equation of the circle for each case. The equation of the circle is a fundamental concept in mathematics, and it has numerous applications in various fields. By understanding the properties of a circle and the given information, we can find the equation of a circle with a given radius and center.

The equation of the circle with a radius of 5, whose center lies on the y-axis and passes through the point (3, 2), is:

x^2 + (y + 2)^2 = 25 or x^2 + (y - 6)^2 = 25

Note: The final answer is not unique, as there are two possible centers: (0, -2) and (0, 6).
Frequently Asked Questions (FAQs) about Finding the Equation of a Circle

In our previous article, we discussed how to find the equation of a circle with a given radius and center. We also explored the process of finding the equation of a circle with a radius of 5, whose center lies on the y-axis and passes through the point (3, 2). In this article, we will answer some frequently asked questions (FAQs) about finding the equation of a circle.

Q: What is the general equation of a circle?

A: The general equation of a circle with center (h, k) and radius r is given by:

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

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

A: To find the center of a circle, you need to know the coordinates of two points on the circle. You can use the distance formula to find the distance between the two points, and then use the midpoint formula to find the center of the circle.

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 equation of a circle with a given radius and center?

A: To find the equation of a circle with a given radius and center, you can use the general equation of a circle:

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

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

Q: What is the significance of the y-axis in finding the equation of a circle?

A: The y-axis plays a crucial role in finding the equation of a circle when the center lies on the y-axis. In this case, the x-coordinate of the center is 0, and the y-coordinate is unknown. You can use the equation of the circle to find the y-coordinate of the center.

Q: How do I find the equation of a circle with a radius of 5, whose center lies on the y-axis and passes through the point (3, 2)?

A: To find the equation of a circle with a radius of 5, whose center lies on the y-axis and passes through the point (3, 2), you can follow these steps:

Write the equation of the circle with center (0, k) and radius 5: (x - 0)^2 + (y - k)^2 = 5^2
Simplify the equation: x^2 + (y - k)^2 = 25
Substitute the coordinates of the point (3, 2) into the equation: (3)^2 + (2 - k)^2 = 25
Solve for k: 2 - k = ±4
Find the possible values of k: k = 2 ± 4
Find the equation of the circle for each possible value of k: x^2 + (y + 2)^2 = 25 or x^2 + (y - 6)^2 = 25

Q: What are the possible centers of a circle with a radius of 5, whose center lies on the y-axis and passes through the point (3, 2)?

A: The possible centers of a circle with a radius of 5, whose center lies on the y-axis and passes through the point (3, 2), are (0, -2) and (0, 6).

Q: What is the equation of a circle with a radius of 5, whose center lies on the y-axis and passes through the point (3, 2)?

A: The equation of a circle with a radius of 5, whose center lies on the y-axis and passes through the point (3, 2), is:

x^2 + (y + 2)^2 = 25 or x^2 + (y - 6)^2 = 25

In this article, we have answered some frequently asked questions (FAQs) about finding the equation of a circle. We have discussed the general equation of a circle, how to find the center of a circle, and how to find the equation of a circle with a given radius and center. We have also explored the process of finding the equation of a circle with a radius of 5, whose center lies on the y-axis and passes through the point (3, 2).