JEE Main & Advanced Mathematics Circle and System of Circles Angle of Intersection of Two Circles

The angle of intersection between two circles \[S=0\] and \[S'=0\] is defined as the angle between their tangents at their point of intersection.      

If \[S\equiv {{x}^{2}}+{{y}^{2}}+2{{g}_{1}}x+2{{f}_{1}}y+{{c}_{1}}=0\]

\[S'\equiv {{x}^{2}}+{{y}^{2}}+2{{g}_{2}}x+2{{f}_{2}}y+{{c}_{2}}=0\]

are two circles with radii \[{{r}_{1}},\,\,{{r}_{2}}\] and d be the distance between their centres then the angle of intersection \[\theta \]  between them is given by \[\cos \theta =\frac{r_{1}^{2}+r_{2}^{2}-{{d}^{2}}}{2{{r}_{1}}{{r}_{2}}}\] or \[\cos \theta =\frac{2\,({{g}_{1}}{{g}_{2}}+{{f}_{1}}{{f}_{2}})-({{c}_{1}}+{{c}_{2}})}{2\sqrt{g_{1}^{2}+f_{1}^{2}-{{c}_{1}}}\,\,\sqrt{g_{2}^{2}+f_{2}^{2}-{{c}_{2}}}}\].

Condition of Orthogonality : If the angle of intersection of the two circles is a right angle \[(\theta ={{90}^{o}})\], then such circles are called orthogonal circles and condition for orthogonality is \[2{{g}_{1}}{{g}_{2}}+2{{f}_{1}}{{f}_{2}}={{c}_{1}}+{{c}_{2}}\].