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

  • question_answer
    If two circles \[{{(x-1)}^{2}}+{{(y-3)}^{2}}={{r}^{2}}\] and \[{{x}^{2}}+{{y}^{2}}-8x+2y+8=0\] intersect in two distinct points, then [IIT 1989; Karnataka CET 2002;                    DCE 2000, 01; AIEEE 2003; MP PET 2004]

    A)            \[2<r<8\]                                 

    B)            \[r=2\]

    C)            \[r<2\]                                      

    D)            \[r>2\]

    Correct Answer: A

    Solution :

               When two circles intersect each other, then                    Difference between their radii<Distance between centers \[\Rightarrow r-3<5\Rightarrow r=8\]                                           .....(i)                    Sum of their radii > Distance between centres    .....(ii)                    \[\Rightarrow \] \[r+3>5\Rightarrow r>2\]                    Hence by (i) and (ii), \[2<r<8\]


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