mathleaks.com mathleaks.com Start chapters home Start History history History expand_more
{{ item.displayTitle }}
navigate_next
No history yet!
Progress & Statistics equalizer Progress expand_more
Student
navigate_next
Teacher
navigate_next
{{ filterOption.label }}
{{ item.displayTitle }}
{{ item.subject.displayTitle }}
arrow_forward
{{ searchError }}
search
{{ courseTrack.displayTitle }}
{{ statistics.percent }}% Sign in to view progress
{{ printedBook.courseTrack.name }} {{ printedBook.name }}

# Drawing Circles in the Coordinate Plane

Rule

## Standard Equation of a Circle

The standard equation of a circle with radius is the relation between the center of the circle, and any point that lies on the circle,

The distance between the center and any point on the circle is always the radius,

The equation of a circle can be derived from the Pythagorean Theorem.

### Derivation

Consider a circle centered at such that the point lies on the circle.

A right triangle can be created by drawing a horizontal segment to the right of and a vertical segment down from

The hypotenuse of the triangle has the length To find the length of the legs, the coordinates of the third vertex are marked.

The lengths of the legs can now be expressed as the difference between the - and -coordinates of the vertices.

Therefore, the length of the legs can be expressed as and Since the triangle is a right triangle, the Pythagorean Theorem applies. Here, and are the lengths of the legs and is the hypotenuse. Substituting the stated values for the triangle drawn above yields the standard equation of a circle.

fullscreen
Exercise

Graph the circle from the equation

Show Solution
Solution

If a circle's center and radius are known the circle can be drawn in a coordinate plane by writing the given equation in standard form. From the given equation, it can be seen that the center of the circle is and its radius is First, the center can be marked.

The circle can now be drawn using a compass. Set the compass to the length of units by placing the needle at the center and the pencil at any point units away from

Now, draw the corresponding circle with the pencil on the compass.

The following graphs shows the circle with the given equation.

Method

## Writing the Standard Equation of a Circle

The standard equation for a circle can be written using the radius, the center, and a point on the circle.

The equation of the circle shown above can be written using the following method.

### 1

State the known values

The general form of the equation of a circle is where is the center of the circle and is the radius. Of the values needed to write the equation, the center is given: Thus, and The radius of the circle remains to be found.

### 2

Find the length of the radius

The radius of the circle can be drawn from to

To determine the radius, calculate the distance between the points and By substituting the coordinates, and into the distance formula, the radius can be calculated.
Evaluate right-hand side
Therefore, the radius of the circle is units.

### 3

Substitute values
The equation can be written by substituting the values of and into the rule.
The equation for the circle is
fullscreen
Exercise

Given a circle with center at and the radius units. Determine if the point lies on the circle.

Show Solution
Solution
To determine if the point lies on the circle, we need to find the equation of the circle. Then, the point lies on the circle if and only if it satisfies the equation. The standard equation of a circle is: where is a point on the circle, is the center and is the radius. By substituting the center of the circle, and its radius, into the rule, the circle's equation can be written. If the point satisfies the equation, it lies on the circle. Before we test the point, the equation can be simplified. Now, we'll substitute the point to see if it satisfies the equation.
Evaluate left-hand side
Because does not equal the point does not satisfy the equation. Thus, the point does not lie on the circle.
Method

## Find the Standard Equation of a Circle by Completing Squares

The equation of a circle is not always given in standard form. Consider the following as an example. The equation can be written in standard form by completing the square.

### 1

Identify binomials

The standard equation of a circle consist of two factored perfect square trinomials on the left-hand side. By completing the square on the - and -terms, perfect square trinomials can be created. The terms on the left-hand side of the equation can be rearranged so that the terms with the same variables are next to each other. It is necessary to complete the square on and

### 2

Find the constants to complete the square
The constants needed to complete the square can be found by splitting the second term in the binomials into two factors, where one is This is shown for the -terms.
By adding the square of to the binomial, a perfect square trinomial is created. The same procedure is done for the -terms, resulting in

### 3

Add the constants to the equation

The constants should now be added to the equation. The terms on the right-hand side can be added.

### 4

Simplify the equation
The equation can be simplified by factoring the perfect square trinomials on the left-hand side. First the trinomial will be factored, then the
Therefore, the equation of the circle, written in standard form, is: The center of the circle can now be identified as and the radius as units.
{{ 'mldesktop-placeholder-grade-tab' | message }}
{{ 'mldesktop-placeholder-grade' | message }} {{ article.displayTitle }}!