The center of a circle is defined as a point inside the circle that is equidistant from all the points on the circumference. It is generally denoted by the coordinates (h, k) and is the point from where all the radii pass. The center of the circle can also be described as the mid-point of any diameter of the circle, as it lies exactly halfway between the endpoint of the diameter.
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.
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 the Centre of a Circle?
There are several ways to find the center of a circle, including using basic geometry, applying the midpoint formula, or examining the equation of the circle in standard form (x − h)2 + (y − k)2 = r2, where (h, k) represents the center's coordinates.
Finding the Center when only the Circle is given:
Step 1: Draw a chord AB in the circle and measure its length (which is 5 inches in this case).
Step 2: Draw another chord CD, parallel to AB, making sure it is also 5 inches long.
Step 3: Use a ruler to connect points C and B with a straight line segment.
Step 4: Similarly, connect points A and D with a straight line.
Step 5: The point where the two lines AD and BC intersect will be the center of the circle.
How to Find a Center of Circle with Two Points?
If two points on the 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 a circle.
Step 2: Locate (a, b) and (c, d), their coordinates.
Step 3: Use the following formula to find their Midpoint: Midpoint (h, k) = [(a + c) / 2, (b + d) / 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 the opposite midpoint of the the 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).
Alternative Method for Finding the Center of a Circle
Alternatively center of any circle is found by,
Examine the 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 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 midpoint of the 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
The 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
The 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.
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 the the circle, given points 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 = (y1 + 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:
Solved Examples of the Center of a Circle
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
Question 1: Determine the circle's equation with radius 4 and center (−2, 3).
Question 2: The circle represented by the equation x2 + y2 - 6x + 8y + 9 = 0 has a center and a radius.
Question 3: Determine the equation of the circle with the midpoint of PQ at its center, given two points, P(2,5) and Q(−3, −1).
Question 4: 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.