The General Form of the equation of a circle is x^2 + y^2 + 2gx +2fy + c = 0. Find the centre of the circle and the radius of the circle in terms of g f and c.
Given a circle in the general form you can complete the square to change it into the standard form.x2 + 2gx + y2 +2fy +c = 0 (1). General form of an equation which has the completing the square method applied to it is (x+d)2 + e. By completing the square we want the expression to look like (x+d)2 + e + (y + j)2 + k + c = 0, where d e j and k are all constants.Expanding this expression: x2 +2dx + d2 + e + y2 + 2jy + j2 + k +c = 0 (2). Comparing equatons (1) and (2) d=g, f=j, d2 + e = j2 + k =0. Therefore e = - g2 and k = -f2. Equation (1) can be rewritten as (x+g)2 + - g2 + (y + f)2 + -f2 + c = 0. Rearranging: (x+g)2 + (y + f)2 = g2 + f2- c. The equation of a circle with centre (a, b) and radius r is (x - a)2 + (y - b)2 = r2. Therefore a = -g, b= -f, r = √(g2 + f2- c).Answer: Centre is (-g, -f), Radius is r = √(g2 + f2- c)
