# Equations of a Circumference: A Comprehensive Guide

## Circumference as Geometric Locus

The geometric locus is a group of points that satisfy certain geometric conditions. The geometric locus of the circumference is the group of points that are at the same distance ** r** from a point called the center.

### Example

Determine the geometric locus that describes a point that moves a distance of 4 units around a fixed point with coordinates (0,0).

#### Steps

- Identify the fixed point on the plane.
- Draw a point at a distance of 4 units to each of the four cardinal points.
- Join the drawn points as a circle at 4 units from the fixed point (0, 0).

#### Answer

The resulting figure is a circumference with center at (0, 0) and radius equal to 4.

## Types of Equations of the Circumference

There are three ways to represent the equation of the circumference:

**Canonical form****Ordinary form****General equation**

For the **canonical form**, the center of the circumference is considered to be at the origin ** C (0, 0)**, the distance is from the center to any of the points of the circumference

*P(x,y)**,*and its radius is a length equal to

**:**

*r*Given points **C(0,0)** and **P(x,y)**, the distance between them will be the length of the radius. Applying the formula of the distance between two points, we have:

By raising both sides of the equation to the square, we have:

This expression is known as the equation of the circumference with its center at the origin, also known as the canonical form.

### Example

Determine the equation of the circumference with its center at the origin and a radius equal to 7.

#### Steps

- If the center of the circumference is at the origin, then the following equation is used:
- Substitute the value of the radius in the previous equation:

#### Answer

Raise the radius to the square to obtain the following expression:

### Example

Determine the equation of the circumference that contains point (2,5) and its center is at the origin C(0,0).

#### Steps

- In this exercise, the length of the radius is not known, it should be determined with the given data and the following equation:
- Once the radius has been established, substitute its value in the canonical form of the equation of the circumference.

#### Answer

The square is canceled with the square root and the following equation is obtained:

The equation of the circumference in its ordinary form has its center outside the origin at ** C(h,k)** and its radius is the distance from the center to any point within the circumference

*P(x,y)**:*

Calculating the distance from the center to the point for obtaining the radius, we have:

Both sides of the equality are raised to the square:

This equation is known as the **equation of the circumference** with its center outside the origin, also known as the ordinary form of the equation of the circumference.

### Example

Determine the equation of the circumference with center at (-1,3) and with radius length equal to 6.

#### Steps

- To find the equation, use the following expression because its center is outside the origin.
- Substitute the center C(-1,3) and the radius in the previous equation:
- Apply the rules of signs and raise to the square:

#### Answer

### Example

Determine the equation of the circumference with center at (-2,-1) that passes through point P(2,4).

#### Steps

- Obtain the value of the radius with the following equation:
- Apply the rules of signs and obtain the square root:
- Knowing the value of the radius and the center C(-2,-1), substitute the values in the following equation:

#### Answer

The resulting equation is:

To obtain the general form of the circumference equation, we start from its ordinary form.

We work with the binomials to the square and the resulting equation is as follows:

Making the equation equal zero, we have:

Substitute to obtain the following:

The previous expression is known as **the general equation of the circumference**.

From the *value* of the radius, we can determine its graphic representation:

- If
**r < 0**, then the equation does not represent a circumference, so it cannot be graphically represented because there are no negative radiuses. - If
**r = 0**, then it represents a point which is represented by the coordinates of the center. - If
**r > 0**, the geometric locus is from a circumference and it is possible to represent it in a plane.

### Example

Given the equation of the circumference on ordinary form, obtain the equation in its general form.

#### Steps

- Expand binomials to the square, as follows:
- Arrange the quadratic terms first, then the linear ones and finally, the equation must equal to zero:

#### Answer

Simplify the expression by adding all independent terms. The resulting equation is:

## Equation of a Circumference that Satisfies Three Conditions

Consider the case in which you have a circumference that passes through three points *A(x _{1},y_{1})*

*,*

**and**

*B(x*_{2},y_{2})**, these three points are substituted in the equation of the circumference in its general form where the missing values are**

*C(x*_{3},y_{3})

*D**,*and

**E****. In this way, you will obtain a system of three equations with three variables to solve.**

*F*You can see this case illustrated in the following example.

### Example

Determine the equation of the circumference that passes through points O(2,-2), P(3,1) and Q(-3,-2).

#### Steps

- Substitute the points in the general formula:
- Evaluate the values of each of the equations and simplify as follows:
- You have 3 equations with three variables. You can use any method you know to solve the system of equations: substitution, addition or elimination.
- Applying the elimination method, you add equations P and Q:
- Equation O is multiplied by 3 and equation Q is multiplied by 2, then both are added:
- Equation R is multiplied by -10 and is added to equation S for then solving for variable F:
- Knowing the value for variable
*F,*substitute in equation S and solve for variable*E*: - Knowing the values
*F*= -12 and*E*= -1, substitute in any of the three original equations,*O, P*or*Q*to obtain the value for*D*.

#### Answer

Substitute the values for *D, E* and *F* in the general equation of the circumference to obtain the following expression:

## Polar Form of the Circumference Equation

To obtain the polar form of the equation of the circumference, you should express it in terms of ** r** and , and substitute and . The following example can help you understand this concept.

### Example

Given the equation of the circumference , obtain its polar form.

#### Steps

- Expand binomials to the square, as follows:
- Substitute and in the resulting expression:
- From the first two terms, take out r
^{2}as a common factor: - Considering the identity =1

substitute it by 1 in the equation and factorize:

#### Answer

The equation in its polar form is: