Equations of the Circumference: Canonical, General & Polar Forms

Posted on Aug 24, 2024 in Mathematics

Circumference as a Geometric Locus

A geometric locus is a set of points that satisfy specific geometric conditions. The geometric locus of the circumference is the set of all points that are equidistant (a distance of r) from a fixed point called the center.

Example: Finding the Geometric Locus

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

Steps

1. Identify the fixed point: The fixed point is (0, 0) on the plane.
2. Draw points: Draw a point 4 units away in each of the four cardinal directions (up, down, left, right) from the fixed point.
3. Connect the points: Join the drawn points to form a circle with a radius of 4 units centered at the fixed point (0, 0).

Answer

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

Types of Equations of the Circumference

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

1. Canonical Form
2. Ordinary Form
3. General Form

Canonical Form

The canonical form considers the center of the circumference to be at the origin, C(0, 0). The distance from the center to any point on the circumference, P(x, y), is equal to the radius, r.

Using the distance formula between two points, C(0, 0) and P(x, y), we get:

Squaring both sides of the equation:

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

Example: Finding the Equation in Canonical Form (Given Radius)

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

Steps

1. Use the canonical form: Since the center is at the origin, use the equation:
2. Substitute the radius: Substitute the value of the radius (7) into the equation:

Answer

Square the radius to obtain the final equation:

Example: Finding the Equation in Canonical Form (Given Point)

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

Steps

1. Determine the radius: Use the distance formula to find the radius:
2. Substitute into canonical form: Substitute the calculated radius into the canonical form of the circumference equation:

Answer

Simplify the equation:

Ordinary Form

The ordinary form of the circumference equation has its center outside the origin at C(h, k), and its radius is the distance from the center to any point on the circumference, P(x, y).

Calculating the distance between the center and a point to obtain the radius:

Squaring both sides of the equation:

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

Example: Finding the Equation in Ordinary Form (Given Radius)

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

Steps

1. Use the ordinary form: Since the center is outside the origin, use the equation:
2. Substitute the center and radius: Substitute the center, C(-1, 3), and the radius (6) into the equation:
3. Apply rules of signs and square: Apply the rules of signs and square the radius:

Answer

The final equation in ordinary form is:

Example: Finding the Equation in Ordinary Form (Given Point)

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

Steps

1. Calculate the radius: Use the distance formula to find the radius:
2. Apply rules of signs and find the square root: Apply the rules of signs and calculate the square root:
3. Substitute into ordinary form: Substitute the calculated radius and the center, C(-2, -1), into the ordinary form equation:

Answer

The resulting equation in ordinary form is:

General Form

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

Expanding the binomials:

Setting the equation equal to zero:

Substituting :

This is the general equation of the circumference.

Interpreting the Radius Value:

1. If r < 0, the equation does not represent a circumference (negative radii are not possible).
2. If r = 0, the equation represents a point, which is the center of the circumference.
3. If r > 0, the equation represents a circumference that can be graphed.

Example: Converting from Ordinary Form to General Form

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

Steps

1. Expand the binomials: Expand the binomials:
2. Rearrange the terms: Arrange the terms with the quadratic terms first, then the linear terms, and finally set the equation equal to zero:

Answer

Simplify the expression by combining the independent terms. The resulting equation in general form is:

Equation of a Circumference Satisfying Three Conditions

If a circumference passes through three points, A(x₁, y₁), B(x₂, y₂), and C(x₃, y₃), these points can be substituted into the general form of the circumference equation, , to solve for the missing values D, E, and F. This will result in a system of three equations with three variables.

Example: Finding the Equation Given Three Points

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

Steps

1. Substitute the points into the general form: Substitute the points into the general form equation:
2. Evaluate and simplify: Evaluate the values in each equation and simplify:
3. Solve the system of equations: Use a method like substitution, addition, or elimination to solve the system of equations:
4. Continue solving: (The remaining steps involve solving the system of equations using the elimination method. The details are omitted for brevity.)

Answer

Substitute the values for D, E, and F into the general equation of the circumference to obtain the final equation:

Polar Form of the Circumference Equation

To obtain the polar form of the circumference equation, express it in terms of r and θ, and substitute x = r cos θ and y = r sin θ.

Example: Converting to Polar Form

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

Steps

1. Expand the binomials: Expand the binomials:
2. Substitute: Substitute x = r cos θ and y = r sin θ into the equation:
3. Factor and simplify: Factor and simplify the equation using the trigonometric identity cos² θ + sin² θ = 1: (Details omitted for brevity.)

Answer

The equation in polar form is: