mathleaks.com mathleaks.com Start chapters home Start History history History expand_more Community
Community expand_more
menu_open Close
{{ filterOption.label }}
{{ item.displayTitle }}
{{ item.subject.displayTitle }}
arrow_forward
No results
{{ searchError }}
search
Expand menu menu_open home
{{ courseTrack.displayTitle }}
{{ statistics.percent }}% Sign in to view progress
{{ printedBook.courseTrack.name }} {{ printedBook.name }}
search Use offline Tools apps
Login account_circle menu_open
Circles With and Without Coordinates

Equation of a Circle

Circles have been previously studied in this course. In this lesson, the equation of a circle on a coordinate plane will be derived. Furthermore, when given the equation of a circle, its center and radius will be found, and its graph will be drawn.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Try your knowledge on these topics.

a By using the Distance Formula, find the distance between points and
points
b Identify the center and find the radius of the circle below.
points
c By completing the square, determine the values of and

d Solve the linear equation.

e Solve the quadratic equation.

Write the smaller solution first.

Challenge

Investigating Points on a Circle's Circumference

In the diagram below, a circle centered at the origin with radius has been drawn. Some points that lie on the circle can be identified.
identify points
Given that is a point that lies on the circle and the coordinate of is , can you determine the coordinate of

Discussion

Deriving the Equation of a Circle at the Origin

The equation of a circle on a coordinate plane can be obtained using the Distance Formula. For example, consider a circle centered at the origin with a radius of An arbitrary point lies on the circle.
circle
Since the radius is the distance between and is also This information can be substituted into the Distance Formula.
Simplify
The equation of the given circle was obtained.

Explore

Standard Equation of a Circle

The result previously obtained can be generalized to find the equation of a circle with a certain center and given radius.

On a coordinate plane, consider a circle with radius and center

standard equation of a circle

The standard equation of the above circle is given below.

Proof

On the diagram below, consider an arbitrary point on a circle with coordinates
standard equation of a circle
As it can be seen in the diagram, the distance between the center and the point is This information can be substituted into the Distance Formula.
Simplify
The equation of a circle with radius and center was obtained by using the Distance Formula.
By using this formula, the standard equation of any circle drawn on a coordinate plane can be written.

Example

Finding the Standard Equation of a Circle

Tearrik has one last problem to solve before going to a BBQ. He needs to find the standard equation of the circle shown below.

A circle on a coordinate plane

Tearrik remembers that the standard equation of a circle is However, he does not remember how to find the values of and Help Tearrik get to the BBQ by finding these values!

Hint

The center of the circle is and its radius

Solution

In the standard equation of a circle, is the center of the circle. Therefore, by noting the coordinates of the center in the diagram, the values of and can be determined. Furthermore, the value of is given by the distance between the center and any point on the circle.
A circle on a coordinate plane
The center of the circle can be seen to have the coordinates Therefore, it can be stated that and It can also be seen that the radius of the circle is By substituting these values into the standard equation of a circle, the equation of the given circle can be obtained.
Simplify

Example

Finding the Center and Radius of a Circle

Just like her classmate, Zain has one last problem to solve before getting to go to the BBQ. She has been asked to identify the center and the radius of the circle whose standard equation is given below. Zain has also been asked to graph the circle on a coordinate plane. Help Zain get to the BBQ!

Hint

The standard equation of a circle is Here, the center of the circle is and its radius is

Solution

The standard equation of a circle is shown below. Here, the center of the circle is and its radius is Therefore, it is convenient to rewrite the given equation to match this format.
Rewrite as and Rewrite as

From the obtained equation, the center of the circle can be identified as and its radius as With this information, the circle can be drawn on a coordinate plane.
circle


Example

Rewriting the Equation of a Circle

Sometimes the equation of a circle needs a significant change to be rewritten as the standard equation of a circle. Typically in those cases, the equation can be rewritten by completing the square.

To be allowed to help design her schools basketball court, Dominika was asked to identify the center and the radius of the circle whose equation is given below. By rewriting the above equation as the standard equation of a circle, identify the center and the radius. If any answer is an irrational number, write its exact value.

Hint

Add a number to the expression so that it becomes a perfect square trinomial.

Solution

To rewrite the given equation as the standard equation of a circle, a method called completing the square can be used. This method consists in writing an expression as a perfect square trinomial so that it can be factored as the square of a binomial. First, the expression will be considered. Ignoring the negative sign, the coefficient of the variable is This coefficient is commonly called Next, the term will be added to and subtracted from the expression. Note that adding and subtracting the same value does not modify the expression. Since the value of can be calculated.
Simplify
Therefore, will be added to and subtracted from Then, the resulting perfect square trinomial will be factored and written as the square of a binomial.
The process of completing the square is now finished. Finally, to obtain the standard equation of the circle, the number will be added to both sides of the equation and will be written as Also, the resulting number on the right-hand side will be expressed as a square.
Simplify
The standard equation of the circle was obtained. The center can be identified as and the radius as
circle


Explore

Rewriting the Equation of One More Circle

This time, Dominika wants to play basketball on Sunday. Her father will be okay with that only if she completes her math homework. To do so, Dominika has to identify the center and the radius of the circle whose equation is given below. By rewriting the above equation as the standard equation of a circle, identify the center and the radius. If any of the answers is an irrational number, write its exact value.

Hint

Complete the square for the and the variable.

Solution

To rewrite the given equation as the standard equation of a circle, the process of completing the square needs to be performed for both variables.

Simplify

The standard equation of the circle was obtained. The center can be identified as and the radius as
circle


Closure

Determining a Point on the Circumference of a Circle

The challenge presented at the beginning of this lesson can be solved by writing the equation of the circle.

On a coordinate plane, a circle centered at the origin with radius was drawn. Also, a point on the circle with coordinate was plotted.

circle and point
By writing the standard equation of the circle, find the coordinate of Write the answer as an exact value.

Hint

The standard equation of a circle is where is the center and the radius.

Solution

Recall the standard equation of a circle. Here, is the center and the radius of the circle. Knowing that the center of the circle drawn in the diagram is and that its radius is its standard equation can be written.
Simplify
Recall that the coordinate of is Therefore, to find its coordinate, this value can be substituted into the equation of the circle.
Solve for
The coordinate of can be either or However, from the diagram it can be observed that is located in Quadrant I, where all values of the variable are positive. Therefore, the coordinate of is
circle and point
{{ 'mldesktop-placeholder-grade-tab' | message }}
{{ 'mldesktop-placeholder-grade' | message }} {{ article.displayTitle }}!
{{ grade.displayTitle }}
{{ exercise.headTitle }}
{{ 'ml-tooltip-premium-exercise' | message }}
{{ 'ml-tooltip-programming-exercise' | message }} {{ 'course' | message }} {{ exercise.course }}
Test
{{ 'ml-heading-exercise' | message }} {{ focusmode.exercise.exerciseName }}
{{ 'ml-btn-previous-exercise' | message }} arrow_back {{ 'ml-btn-next-exercise' | message }} arrow_forward