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