Cambridge O Level Additional Mathematics · Syllabus 4037 · Coordinate Geometry of the Circle
General Form of a Circle Equation
What is General Form of a Circle Equation?
The expanded form in which a circle equation is usually presented, written as x squared plus y squared plus 2gx plus 2fy plus c equals 0. It is what the standard form becomes when the brackets are multiplied out, which is why the coefficients of x squared and y squared are equal and there is no xy term. Completing the square in x and in y converts it back, giving centre at minus g comma minus f and radius equal to the square root of g squared plus f squared minus c. The quantity under that square root can be positive, zero or negative, and the three cases are genuinely different: a positive value gives an ordinary circle, a value of zero gives a single degenerate point rather than a circle, and a negative value means no real points satisfy the equation at all.
This definition is part of the Coordinate Geometry of the Circle chapter in Cambridge O Level Additional Mathematics.
General Form of a Circle Equation in context
The general form of a circle equation, \(x^2+y^2+2gx+2fy+c=0\), is the standard form multiplied out, so completing the square in \(x\) and in \(y\) recovers centre \((-g,-f)\) and radius \(r=\sqrt{g^2+f^2-c}\). The value under that root decides what the equation represents: positive gives a real circle, zero gives a single point, and negative gives no real locus at all. The same idea of substituting and checking runs through the rest of the chapter, from classifying a line against a circle by its discriminant to finding where two circles touch along the line joining their centres.