# Results and Formulas of Circle

01. Equation of the circles having the center at $O\left( {0,0} \right)$ and radius $r$ is

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

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

04. Equation of the 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 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

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