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JEE Main & Advanced Mathematics Circle and System of Circles Question Bank System of circles

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    The centre of the circle, which cuts orthogonally each of the three circles \[{{x}^{2}}+{{y}^{2}}+2x+17y+4=0,\] \[{{x}^{2}}+{{y}^{2}}+7x+6y+11=0,\] \[{{x}^{2}}+{{y}^{2}}-x+22y+3=0\] is [MP PET 2003]

    A)            (3, 2)                                         

    B)            (1, 2)

    C)            (2, 3)                                         

    D)            (0, 2) 

    Correct Answer: A

    Solution :

               Let circle \[{{x}^{2}}+{{y}^{2}}+2gx+2fy+c=0\]   .....(i)                    Circle (i) cuts orthogonally each of the given three circles.                    Then according to the condition                    \[2{{g}_{1}}{{g}_{2}}+2{{f}_{1}}{{f}_{2}}={{c}_{1}}+{{c}_{2}}\],                    \[2g+17f=c+4\]                                              .....(ii)                    \[7g+6f=c+11\]                                              .....(iii)                    \[-g+22f=c+3\]                                               .....(iv)                    From (ii), (iii) and (iv), \[g=-3,\,\]\[f=-2\]                    Therefore, the centre of the circle \[(-g,\,-f)=\,(3,\,2)\].


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