JEE Main & Advanced Mathematics Circle and System of Circles Common Chord of Two Circles

(1) Definition : The chord joining the points of intersection of two given circles is called their common chord.

(2) Equation of common chord : The equation of the common chord of two circles

\[{{S}_{1}}\equiv {{x}^{2}}+{{y}^{2}}+2{{g}_{1}}x+2{{f}_{1}}y+{{c}_{1}}=0\]   ….(i)

and    \[{{S}_{2}}\equiv {{x}^{2}}+{{y}^{2}}+2{{g}_{2}}x+2{{f}_{2}}y+{{c}_{2}}=0\]                        ….(ii)

is \[2x\,({{g}_{1}}-{{g}_{2}})+2y\,({{f}_{1}}-{{f}_{2}})+{{c}_{1}}-{{c}_{2}}=0\] i.e., \[{{S}_{1}}-{{S}_{2}}=0\].

(3) Length of the common chord :

\[PQ=2\,(PM)=2\,\sqrt{{{C}_{1}}{{P}^{2}}-{{C}_{1}}{{M}^{2}}}\]

Where \[{{C}_{1}}P=\] radius of the circle \[{{S}_{1}}=0\] and \[{{C}_{1}}M=\] length of the perpendicular from the centre \[{{C}_{1}}\] to the common chord PQ.