# 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 ${S_1} + k{S_2} = 0,\,\,\,k \in \mathbb{R},\,\,k \ne – 1$