Center of a Circle is defined as a point inside the circle that is equidistant from all the points on the circumference of the circle. It is generally denoted using (h, k) points and is the point from where all the radius passes. Cente of the circle is defined as the mid-point of the end point of the diameter of the circle.

In this article, we will learn about, center of a circle, its formulas, and examples in detail.

What is Centre of a Circle?

The fixed point that is equally spaced from every other point on the circle’s perimeter is known as the circle’s center. The radius is the length of the circle that separates any point from the center. The letters “O” or “C” stand for the center of a circle.

Center of Circle Formula

Formula used to find the center of a circle is,

Midpoint (h, k) = [(x₁ + x₂)/2, (y₁ + y₂)/2]

Where any two points on the circle are represented by the points (x₁, y₁) and (x₂, y₂). We must locate the midpoint between two points on the circle in order to determine the coordinates of the center.

How to Find Centre of a Circle?

To find the center of any circle follow the steps added below,

Step 1: Determine Two Points on Circumference of Circle

Select two locations within the circle. These could be any two different points on the circumference, the ends of a diameter, or points along a secant or chord.

Step 2: Use Midpoint Formula

To find the coordinates of the midpoint of the line segment connecting the two points, use the midpoint formula. The formula for finding the midpoint is Midpoint=(x₁+x₂/2 ,y₁+y₂/2), where the coordinates of the two points that have been identified are (x₁, y₁) and (x₂, y₂).

Step 3: Midpoint Coordinates represents Center of Circle

Circle’s center is represented by the coordinates found using the midpoint formula. Let us say these coordinates are (h, k), where k is the center’s y-coordinate and h is its x-coordinate.

Step 4: Check for Further Points

You can verify that the computed center coordinates are equally spaced from each of the additional points on the circle if you have more than one. This attests to the fact that you have located the right center.

Alternative Method for Finding Center of a Circle

Alternatively center of any circle is found by,

Examine Equation of Circle

When the circle’s equation is expressed in the standard form (x – h)² + (y – k)² = r², the center’s coordinates are (h, k), and the radius is denoted by r.

How to Find Center of Circle with Two Points?

If two points on Circumference of a Circle are given then its center is only found when these two points are on opposite segments of the circle. To find the center then follow the steps added below,

Step 1: Identify two points (A and B) on the circle that are on opposite segments of circle.

Step 2: Locate (x₁, y₁) and (x₂, y₂), their coordinates.

Step 3: Use the following formula to find their midpoint: Midpoint (h, k) = [(x₁ + x₂) / 2, (y₁ + y₂) / 2].

Step 4: The midpoint (h, k) is the center of the circle that goes through these two points.

Step 5: Check that the distance between your response and points A and B is equal.

Example: Find the center of a circle, for instance, that passes through (3, 4) and (-3, -4) that are on opposite segment of circle.

Solution:

Let center of circle is (h, k)

(h, k) is the mid-point of any two points on circumference of circle in opposite ,

(h, k) = [(-3 + 3) / 2, (4 – 4) / 2]

(h, k) = (0, 0)

(0, 0) represents the center of this circle that goes through (-3, 4) and (3, -4).

How to Express Center of Circle?

If you are given an equation for a circle or just two points on it, there are multiple ways to determine its center. Some of the common ways are,

Using Chords

Line segments that join two points on a circle are called chords. A chord is referred to as a diameter if it runs through the center of the circle. The circle’s center is the midpoint of a diameter. Given the endpoints of a chord (diameter), (x₁, y₁) and (x₂, y₂) the formula to find the mid point of chord is,

(x₁ + x₂/2, y₁ + y₂/2) is the midpoint.

Using of Secant

A line that crosses a circle twice is called a secant of a circle. The line segment that results when a secant passes through the circle’s center is called a diameter. The center of the circle is the midpoint of the diameter, just like in the chord case.

Using Overlapping Circles

Center of the given circle can be found when working with overlapping circles by using the point of intersection of their common chord, which is a line connecting two points on a circle.

Using Tangents

Radius at the tangency point of a circle is perpendicular to the tangent line. Therefore, the center of the circle can be found by measuring the midpoint of the line segment that connects the tangent line and the point of tangency to the center.

When Equation of Circle is Given to Us

A circle’s standard equation is (x – h)²+(y – k)² = r², where the radius is denoted by r and the circle’s center by (h, k). Therefore, you can determine the values of h and k, which stand for the center’s coordinates, from the provided equation.

Center of Circle Using Midpoint Formula

If we know the coordinates of any two points on the circle that are on other sector of the circle, now we can use the midpoint formula to find the circle’s center. The formula for the midpoint is:

[(x₁ + x₂)/2, (y₁ + y₂)/2] is the midpoint (h, k)

Example: On circle, given points are A(3,4) and B(7, 8) find its center.

Solution:

Suppose we have A(3, 4) and B(7, 8), two points on the circle.

Step 1: Determine the Circle’s Two Points

Select two locations within the circle. In this instance, our selected points are A and B.

Step 2: Use the Midpoint Calculation

To determine the midpoint’s coordinates (h, k), use the following formula:

h = (x₁ ​+ x₂)​/2

k = (y_1​ + y₂)/2​​

Enter A and B’s coordinates into the formula:

h = (3 + 7)/2 = 5

k = (4 + 8)/2 = 6

So, the midpoint (h, k) is (5, 6).

Step 3: Center is represented by midpoint coordinates

Circle’s center is represented by the coordinates (5, 6) that were found using the midpoint formula.

Step 4: Double-check using the additional points (optional)

You can verify that (5,6) is equally spaced from any additional points on the circle if there are any available points.

Step 5: Examine the Circle Equation (Alternative Method)

If the circle’s equation is known, using the formula should also produce (5, 6) as the center.

Therefore, in this case, (5, 6) is the center of the circle formed by points A(3, 4) and B(7, 8).

Related Read:

• Circle
• Area of Circle
• Chords of Circle
• Segment of Circle
• Sector of Circle

Center of a Circle Solved Examples

Example 1: The task at hand is to determine the radius and center of the circle that is shown by the equation (x−3)² + (y+2)² = 25.

Solution:

Given,

(x−3)² + (y+2)² = 25

By comparing with the standard form (x − h)² + (y − k)² = r²

We determine radius and the center is at,

r = 5

(h, k) = (3, −2)

Example 2: Find the radius and center of the circle that passes through the three points A(1,2), B(5,6), and C(−3,4).

Solution:

Given three non-collinear points, use the formula for the center of a circle:

where h = (x₁ + x₂ + x₃)/3 and k = (y₁ + y₂ + y₃)/3

h = (1 + 5 − 3)/3 = 1

k = (2 + 6 + 4)/3 = 4

Center is at (h, k) = (1, 4).

Use distance formula between center and any of provided points to find the radius.

Example 3: Given the Circle’s Equation: (x − 3)² + (y + 4)² = 25.

Solution:

Let us take the case where we have the circle equation (x − 3)² + (y + 4)² = 25.

First, determine center coordinates.

Equation allows us to determine center coordinates (h,k) directly:

• h = 3
• k =−4

Thus, (3, -4) is circle’s center.

Center of Circle Practice Questions

Q1: Determine the circle’s equation with radius 4 and center (−2, 3).

Q2: The circle represented by the equation x² + y² – 6x + 8y + 9 = 0 has a center and a radius.

Q3: Determine the equation of the circle with the midpoint of PQ at its center, given two points, P(2,5) and Q(−3, −1).

Q4: If the circle x² + y² – 2x + 4y – 13 = 0 finds the new center and radius of the translated circle after being translated three units to the right and two units upward.

Center of Circle Frequently Asked Questions

What is Center of Circle Definition?

A circle’s center is the point in the plane that is equally spaced from each of the circle’s points. This is the geometric center of the circle, and the usual form of the circle equation frequently indicates its coordinates as (h, k).

What is a Circle in Mathematics?

A circle is a completely round, two-dimensional figure in mathematics. It is described as the collection of all points on a plane that are equally spaced apart from the center, a fixed point.

What is Circumference of a Circle?

A circle’s circumference is equal to the entire length of its border. The formula for calculating it is C = 2πr.

What is Equation for a Circle?

(x – h)² + (y – k)² = r² is equation for a circle in the coordinate plane, where “r” is the radius and (h, k) denotes the center’s coordinates.

What is Middle of a Circle Called?

Middle of a circle called is Center of a Circle.