Geometry and Measurement — Equation of a circle

About this lesson

This unit teaches students how to write and interpret the equation of a circle in a Cartesian coordinate plane. It begins by defining a circle as a set of points equidistant from a fixed center, and introduces the standard form of the equation (x-d)² + (y-h)² = r², where (d, h) is the center and r is the radius. Students learn to write the equation given the center and radius or diameter, and to find the center and radius from a given equation. The unit also covers determining whether a point lies on, inside, or outside a circle by substituting its coordinates into the equation. Examples and exercises involve circles with centers at the origin, on axes, or in quadrants, and circles that pass through given points or touch lines. The unit then introduces the general form of the circle equation: x² + y² + 2Lx + 2ky + C = 0, and shows how to convert between the standard and general forms. Students practice identifying whether a given second-degree equation represents a circle by checking that coefficients of x² and y² are equal and unity, and that there is no xy term. They also learn to find the center and radius from the general form using the relationships center = (-L, -k) and radius = sqrt(L² + k² - C), provided L² + k² - C > 0. The unit includes exercises on writing the general form from given conditions, such as endpoints of a diameter, and on determining if two circles are congruent by comparing radii. By the end, students should be able to write the equation of a circle in both standard and general forms, extract center and radius from either form, and solve related problems involving points on or relative to circles.