Cartesian equation of a circle — A/L Combined Mathematics

A circle is the locus of a point P(x, y) moving in a plane at a constant distance r (radius) from a fixed point C(a, b) (centre). By the distance formula, (x - a)² + (y - b)² = r². Expanding yields the general Cartesian equation: x² + y² + 2gx + 2fy + c = 0, where the centre is (-g, -f) and the radius is r = √(g² + f² - c). For this equation to represent a real circle, the condition g² + f² - c > 0 must hold; if g² + f² - c = 0, it represents a point circle (-g, -f); if g² + f² - c < 0, it represents an imaginary circle with no real locus.

Check before you answer (A/L Pure Maths Paper 1)

1. General Equation Coefficients: Ensuring the coefficients of x² and y² are equal and non-zero, and the xy cross-term coefficient is strictly zero.
2. Three Non-Collinear Points: Determining unique values of g, f, and c by substituting three given points into x² + y² + 2gx + 2fy + c = 0 and solving the linear system.
3. Diametric Form: Applying (x - x₁)(x - x₂) + (y - y₁)(y - y₂) = 0 directly when given endpoint coordinates of a diameter.