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

## 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.

# Plotting a Circle on a Coordinate Plane

- 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.

# Solving Problems Involving the Equation of the Circle

- 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).