Results and Formulas of a Circle

01. The equation of a circle having the center at O\left( {0,0} \right) and radius r is

{x^2} + {y^2} = {r^2}

02. The equation of a circle having the center \left( {h,k} \right) and radius r is

{\left( {x - h} \right)^2} + {\left( {y - k} \right)^2} = {r^2}

03.  The equation of a circle in general form is {x^2} + {y^2} + 2gx + 2fy + c = 0, whose center is at \left( { - g, - f} \right) and radius is

r = \sqrt {{g^2} + {f^2} - c}

04. The equation of a circle passing through the point of intersection of the circles {S_1} = 0 and {S_2} = 0 is {S_1} + k{S_2} = 0,\,\,\,k \in \mathbb{R}.

05. If {S_1} = 0 and {S_2} = 0 are the equations of two intersecting circles, then {S_1} - {S_2} = 0 is the equation of the common chord.

06. If {S_1} = 0 and {S_2} = 0 are the equations of two circles such that they touch each other, then {S_1} - {S_2} = 0 is the equation of the common tangent.

07. If {S_1} = 0 and {S_2} = 0 are the equations of two non-intersecting circles, then {S_1} - {S_2} = 0 is the equation of the radical axis.

08. The length of the tangent segment from a point \left( {{x_1},{y_1}} \right) to the circle {x^2} + {y^2} + 2gx + 2fy + c = 0 is given by

\sqrt {{x_1}^2 + {y_1}^2 + 2g{x_1} + 2f{y_1} + c}

09. A system of circles coaxal with the circles  {S_1} = 0 and {S_2} = 0 is

{S_1} + k{S_2} = 0,\,\,\,k \in \mathbb{R},\,\,k \ne - 1