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The Circle

Author: Sophia
PDF Version

what's covered
In this lesson, you will analyze the equation and graph of a circle. Specifically, this lesson will cover:

Table of Contents

1. Exploring Graphical Aspects of the Equation of a Circle

A circle is obtained by cutting a plane horizontally through a cone, as shown in the figure.

A 3D representation of a double cone with a horizontal plane intersecting the upper cone perpendicular to its axis, forming a circle. A vertical line runs through the center of the circle, representing the cone’s axis of symmetry.

In terms of distance, a circle is defined as the set of points open parentheses x comma space y close parentheses whose distances from a fixed point C are equal. Consider the circle shown below.

A graph has a circle with its center at the marked point labeled ‘C (h, k)’ in the first quadrant. A dashed line slants upward from the center to a marked point on the circle labeled ‘(x, y).’ The distance between the points C (h, k) and (x, y) is labeled ‘r.’, indicating that this is the radius of the circle.

  • The center of the circle is labeled open parentheses h comma space k close parentheses.
  • open parentheses x comma space y close parentheses represents any point on the circle.
  • The radius of the circle is r.
To get an equation for all points open parentheses x comma space y close parentheses that are on the circle, note that the distance between open parentheses x comma space y close parentheses and open parentheses h comma space k close parentheses is the radius, r.

r equals square root of open parentheses x minus h close parentheses squared plus open parentheses y minus k close parentheses squared end root Apply the distance formula: d equals square root of open parentheses x subscript 2 minus x subscript 1 close parentheses squared plus open parentheses y subscript 2 minus y subscript 1 close parentheses squared end root with d equals r comma open parentheses x subscript 1 comma space y subscript 1 close parentheses equals open parentheses h comma space k close parentheses comma and open parentheses x subscript 2 comma space y subscript 2 close parentheses equals open parentheses x comma space y close parentheses.
r squared equals open parentheses x minus h close parentheses squared plus open parentheses y minus k close parentheses squared Square both sides of the equation.

By writing the equation with r squared on the other side, we obtain the standard form of the equation of a circle, as shown in the formula below.

formula to know
Standard Form Equation of a Circle
open parentheses x minus h close parentheses squared plus open parentheses y minus k close parentheses squared equals r squared comma where open parentheses x comma space y close parentheses is any point on the circle, open parentheses h comma space k close parentheses is the center of the circle, and r is the radius.

EXAMPLE

A circle has equation open parentheses x plus 1 close parentheses squared plus open parentheses y minus 3 close parentheses squared equals 5.

By matching this with the standard form open parentheses x minus h close parentheses squared plus open parentheses y minus k close parentheses squared equals r squared comma we see that h equals short dash 1 comma k equals 3 comma and r squared equals 5 comma which means r equals square root of 5.

Then, this circle has the center located at open parentheses short dash 1 comma space 3 close parentheses and has radius square root of 5.

try it
A circle has equation open parentheses x minus 5 close parentheses squared plus open parentheses y plus 6 close parentheses squared equals 48.
What are the center and radius of the circle?
Comparing to the standard form open parentheses x minus h close parentheses squared plus open parentheses y minus k close parentheses squared equals r squared comma we see that h equals 5 comma k equals short dash 6 comma and r equals square root of 48 comma which can be written in simplest form as 4 square root of 3.

Thus, the center is located at open parentheses 5 comma space short dash 6 close parentheses and the radius is 4 square root of 3.

The equation open parentheses x plus 1 close parentheses squared plus open parentheses y minus 3 close parentheses squared equals 5 makes it very simple to identify the circle’s center and radius.

If we were to expand the expression on the left and collect like terms and write an equivalent equation set to zero, we get another form called the general form.

x squared plus 2 x plus 1 plus y squared minus 6 y plus 9 equals 5 Expand the square terms.
x squared plus 2 x plus y squared minus 6 y plus 10 equals 5 Combine like terms on the left side.
x squared plus 2 x plus y squared minus 6 y plus 5 equals 0 Subtract 5 from both sides.

The equation x squared plus 2 x plus y squared minus 6 y plus 5 equals 0 is an example of the general form of the equation of a circle.

When a circle’s equation is in general form, completing the square is required to identify its center and radius.

EXAMPLE

Consider the equation 3 x squared minus 12 x plus 3 y squared plus 18 y minus 36 equals 0.

Complete the square and identify its center and radius.

3 x squared minus 12 x plus 3 y squared plus 18 y minus 36 equals 0 This is the original equation.
x squared minus 4 x plus y squared plus 6 y minus 12 equals 0 Since x squared and y squared both have coefficient 3, divide both sides by 3. Remember that completing the square can only be done if the coefficients of the square terms are 1.
x squared minus 4 x plus y squared plus 6 y equals 12 Add 12 to both sides.
x squared minus 4 x plus 4 plus y squared plus 6 y plus 9 equals 12 plus 4 plus 9 1 half the coefficient of x is -2, then open parentheses short dash 2 close parentheses squared equals 4.
1 half the coefficient of y is 3, then 3 squared equals 9.
open parentheses x minus 2 close parentheses squared plus open parentheses y plus 3 close parentheses squared equals 25 Rewrite each trinomial as the square of a binomial. Simplify the right side.

From the equation, we can see now that the center of the circle is open parentheses 2 comma space short dash 3 close parentheses and it has radius 5.

try it
Consider the circle whose equation is x squared plus y squared plus 10 y equals 0.
Write the equation in standard form.
The equation already has all variable terms to one side. Note that there is only one x-term, so we only need to complete the square on the y terms:

x squared plus y squared plus 10 y plus 25 equals 0 plus 25 Add 25 to both sides.
Note: 1 half open parentheses 10 close parentheses equals 5 comma then 5 squared equals 25.
x squared plus open parentheses y plus 5 close parentheses squared equals 25 Write y squared plus 10 y plus 25 as a binomial squared, then simplify the right side.

The standard form of the equation is x squared plus open parentheses y plus 5 close parentheses squared equals 25.
Identify the center and radius of the circle.
Comparing to the equation open parentheses x minus h close parentheses squared plus open parentheses y minus k close parentheses squared equals r squared comma we see that h equals 0 comma k equals short dash 5 comma and r equals square root of 25 equals 5. Thus, the center is located at open parentheses 0 comma space short dash 5 close parentheses and the radius is 5.

2. Writing the Equation of a Circle From Given Information

Given information about a circle, we can write its equation in standard form.

EXAMPLE

Write the equation of a circle whose center is open parentheses 0 comma space short dash 2 close parentheses and which has radius 3.

The center is open parentheses h comma space k close parentheses equals open parentheses 0 comma space short dash 2 close parentheses and its radius is r equals 3.

Substituting into the standard form, we have open parentheses x minus 0 close parentheses squared plus open parentheses y minus open parentheses short dash 2 close parentheses close parentheses squared equals 3 squared.

Simplifying, the equation is written x squared plus open parentheses y plus 2 close parentheses squared equals 9.

The graph of the circle is shown.

A graph has a circle with its center at (0, −2) and a radius of 3 units. There is a marked point on the circle labeled ‘(x, y)’ located in the first quadrant. A dashed line slants upward from the point (0, −2) to the point (x, y).

try it
A circle has diameter 18 and center open parentheses short dash 1 comma space 4 close parentheses.
Write the equation of the circle in standard form.
Note that the diameter is 18, but the standard form uses the radius. Find the radius by taking half the diameter. Thus, r equals 9.

We know the center is open parentheses short dash 1 comma space 4 close parentheses comma which means h equals short dash 1 and k equals 4.

Now we are ready to substitute into the standard equation, open parentheses x minus h close parentheses squared plus open parentheses y minus k close parentheses squared equals r squared.

We have open parentheses x minus open parentheses short dash 1 close parentheses close parentheses squared plus open parentheses y minus 4 close parentheses squared equals 9 squared comma or in simplest form, open parentheses x plus 1 close parentheses squared plus open parentheses y minus 4 close parentheses squared equals 81.

EXAMPLE

A circle has a diameter with endpoints open parentheses short dash 1 comma space short dash 8 close parentheses and open parentheses 3 comma space 4 close parentheses comma as pictured below.

A graph with a circle centered at (1, −2) and a radius of approximately 6 units. A dashed line slants upward from the marked point at (−1, −8) to the marked point at (3, 4), both on the circle.

To write the equation of a circle, we need its center and radius.

The center is the midpoint of the line segment that joins open parentheses short dash 1 comma space short dash 8 close parentheses and open parentheses 3 comma space 4 close parentheses.

open parentheses x subscript m comma space y subscript m close parentheses equals open parentheses fraction numerator x subscript 1 plus x subscript 2 over denominator 2 end fraction comma space fraction numerator y subscript 1 plus y subscript 2 over denominator 2 end fraction close parentheses This is the formula for the midpoint.
equals open parentheses fraction numerator short dash 1 plus 3 over denominator 2 end fraction comma space fraction numerator short dash 8 plus 4 over denominator 2 end fraction close parentheses Substitute open parentheses x subscript 1 comma space y subscript 1 close parentheses equals open parentheses short dash 1 comma space short dash 8 close parentheses and open parentheses x subscript 2 comma space y subscript 2 close parentheses equals open parentheses 3 comma space 4 close parentheses.
equals open parentheses 1 comma space short dash 2 close parentheses Simplify.

Therefore, the center is open parentheses 1 comma space short dash 2 close parentheses.

There are two options to find the radius:

  • Find the distance from the center to one of the endpoints.
  • Find the distance between the endpoints, which is the diameter. Then, divide the diameter by 2.
Using the first option, the radius is the distance between the points open parentheses 1 comma space short dash 2 close parentheses and open parentheses 3 comma space 4 close parentheses.

r equals square root of open parentheses x subscript 2 minus x subscript 1 close parentheses squared plus open parentheses y subscript 2 minus y subscript 1 close parentheses squared end root This is the distance formula, and we are finding the radius.
r equals square root of open parentheses 3 minus 1 close parentheses squared plus open parentheses 4 minus open parentheses short dash 2 close parentheses close parentheses squared end root Substitute open parentheses x subscript 1 comma space y subscript 1 close parentheses equals open parentheses 1 comma space short dash 2 close parentheses and open parentheses x subscript 2 comma space y subscript 2 close parentheses equals open parentheses 3 comma space 4 close parentheses.
r equals square root of 2 squared plus 6 squared end root Simplify the squared expressions.
r equals square root of 40 Simplify under the radical.

Ordinarily we would simplify the radical, but since it is getting substituted into the equation open parentheses x minus h close parentheses squared plus open parentheses y minus k close parentheses squared equals r squared comma it is best kept as a single radical for now.

Now, find the equation of the circle.

open parentheses x minus h close parentheses squared plus open parentheses y minus k close parentheses squared equals r squared This is the standard form of the equation of a circle.
open parentheses x minus 1 close parentheses squared plus open parentheses y minus open parentheses short dash 2 close parentheses close parentheses squared equals open parentheses square root of 40 close parentheses squared Replace open parentheses h comma space k close parentheses equals open parentheses 1 comma space short dash 2 close parentheses and r equals square root of 40.
open parentheses x minus 1 close parentheses squared plus open parentheses y plus 2 close parentheses squared equals 40 Simplify.

The equation of the circle is open parentheses x minus 1 close parentheses squared plus open parentheses y plus 2 close parentheses squared equals 40.

try it
A circle has center open parentheses 2 comma space 5 close parentheses and contains the point open parentheses 8 comma space short dash 1 close parentheses.
Find the radius of the circle. Leave your answer as an unsimplified radical.
The radius is the distance between open parentheses 2 comma space 5 close parentheses and open parentheses 8 comma space short dash 1 close parentheses.

d equals square root of open parentheses x subscript 2 minus x subscript 1 close parentheses squared plus open parentheses y subscript 2 minus y subscript 1 close parentheses squared end root This is the distance formula.
d equals square root of open parentheses 8 minus 2 close parentheses squared plus open parentheses short dash 1 minus 5 close parentheses squared end root Replace open parentheses x subscript 1 comma space y subscript 1 close parentheses with open parentheses 2 comma space 5 close parentheses and open parentheses x subscript 2 comma space y subscript 2 close parentheses with open parentheses 8 comma space short dash 1 close parentheses.
d equals square root of 6 squared plus open parentheses short dash 6 close parentheses squared end root Simplify within parentheses.
d equals square root of 72 Simplify under the radical.

While we could simplify this further, it is easier to work with the answer in this form in the next question. Therefore, the radius is r equals square root of 72.
Write the equation of the circle in standard form.
Using h equals 2 comma k equals 5 comma and r equals square root of 72 comma the standard form of the equation is open parentheses x minus 2 close parentheses squared plus open parentheses y minus 5 close parentheses squared equals open parentheses square root of 72 close parentheses squared comma or in simplest form, open parentheses x minus 2 close parentheses squared plus open parentheses y minus 5 close parentheses squared equals 72.

summary
In this lesson, you explored the graphical aspects of the equation of a circle, which is a collection of points whose distances from one fixed point are equal. The key characteristics of a circle are its center and radius. You also learned that given a circle's equation, you can analyze these characteristics. In addition, given information about a circle—such as a circle's center and radius—you are able to write the equation of a circle.

SOURCE: THIS TUTORIAL HAS BEEN ADAPTED FROM OPENSTAX "PRECALCULUS” BY JAY ABRAMSON. ACCESS FOR FREE AT OPENSTAX.ORG/DETAILS/BOOKS/PRECALCULUS-2E. LICENSE: CREATIVE COMMONS ATTRIBUTION 4.0 INTERNATIONAL.

Formulas to Know
Standard Form Equation of a Circle

open parentheses x minus h close parentheses squared plus open parentheses y minus k close parentheses squared equals r squared comma where open parentheses x comma space y close parentheses is any point on the circle, open parentheses h comma space k close parentheses is the center of the circle, and r is the radius.

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