question_answer

The length  of the common chord of the circles \[{{x}^{2}}+{{y}^{2}}+2x+3y+1=0\]and \[{{x}^{2}}+{{y}^{2}}+4x+3y+2=0\]is  [MP PET 2000]

A)            \[9/2\]                                        

B)            \[2\sqrt{2}\]

C)            \[3\sqrt{2}\]                             

D)            \[3/2\]

Correct Answer: B

Solution :

           Equation of common chord PQ is \[2x+1=0\]                    \[{{C}_{1}}M\] = perpendicular distance of common chord from centre \[{{C}_{1}}\]\[=\left| \frac{-2+1}{\sqrt{{{2}^{2}}}} \right|=\left| -\frac{1}{2} \right|\]                    Here; \[{{C}_{1}}\,\left( -1,-\frac{3}{2} \right)\,,\] \[{{r}_{1}}=\frac{3}{2}={{C}_{1}}P\]                    PQ = 2PM \[=2\sqrt{{{C}_{1}}{{P}^{2}}-{{C}_{1}}{{M}^{2}}}\]\[=2\sqrt{\frac{9}{4}-\frac{1}{4}}=2\sqrt{2}\].