# 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.

By squaring the equation, the following is obtained:

If it is developed:

And these changes are made:

Another way to write the equation is obtained:

The center is:

The radius fulfills the relation:

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

1. The coefficients of x^{2} and y^{2} 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.

### 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:

#### Examples

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

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

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 for the coordinates of the points, the following system is obtained:

Determine whether the equation 4x^{2} + 4y^{2} − 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 x^{2} and y^{2} are different from 1, divide by 4:

2. There is no term in xy.

3.

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