The Coordinate Geometry of Circles: (x - a)^2 + (y - b)^2 = r^2

Understanding the General Equation of a Circle

The equation of a circle with its centre at origin (0,0) is represented as x² + y² = r² where r is the radius of the circle.
The equation for a circle with a center not at the origin, (a, b), is (x - a)² + (y - b)² = r².
Here, (a, b) represents the coordinates of the center of the circle, and r represents the radius.

To plot the circle, start by locating the circle’s center (a, b) on the coordinate plane.
Next, plot a point that lies r-units away from the center to represent a point on the edge of the circle. To achieve an accurate circle, it might be helpful to plot multiple points.
Sketch the circle through these points ensuring all points are the same distance (r) from the centre.
Remember, every point on the circle is r units away from the center, and lies within the region defined by the equation.

Substituting any (x, y) coordinate pair into the equation of a circle should yield a true statement if the point lies on the circle.
Manipulating the equation (x - a)² + (y - b)² = r² can enable you to find the radius, or the coordinates of the circle’s center, given other pieces of information.
Many problems can be solved by substituting known values into the formula or by rearranging the formula to solve for the unknown variable.
Real-world applications can involve ranking circles by size (r), or distance from the origin (a, b).