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).