Category : JEE Main & Advanced
(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.
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