Equation of a Circle

The equation of a circle is the locus of points on the plane that are equidistant from a fixed point called the center.

Equation of a Circle

Equation of a Circle

Equation of a Circle

By squaring the equation, the following is obtained:

Equation of a Circle

If it is developed:

Developed Circle Equation

And these changes are made:

Circle Operations

Another way to write the equation is obtained:

Circle Operations

The center is:

Center of a Circle

The radius fulfills the relation:

Radius Formula


An equation of the type: Equation Type can be a circle if:

1. The coefficients of x2 and y2 are 1 or if they both have the same coefficient that does not equal 1, all terms of the equation can be divided by the value of the coefficient.

2. There is no term in xy.

3. formulas


A Circle with the Origin as its Center

If the center of the circle coincides with the origin of the graph, the equation is reduced to:

Circle at the Origin

Examples

Determine the equation of the circle with its center at point (3, 4) and a radius of 2.

Circle Example

Circle Calculations

Circle Solution


Given the equation of the circle x2 + y2 − 2x + 4y − 4 = 0, find the center and its radius.

Circle Calculations

Circle Solution


Find the equation of the circle that passes through the points A = (2, 0), B = (2, 3), C = (1, 3).

Substituting x and y in the equation Equation Type for the coordinates of the points, the following system is obtained:

System of Equations

Circle Operations

Circle Solution


Determine whether the equation 4x2 + 4y2 − 4x − 8y − 11 = 0 corresponds to a circle, and if so, calculate its center and specify the length of its radius.

1. Since the coefficients of x2 and y2 are different from 1, divide by 4:

formulas

2. There is no term in xy.

3. Circle Operations

Since all three conditions are met, it is a circle.

Circle Calculations

Circle Solution