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

  • question_answer
    If the circles \[{{x}^{2}}+{{y}^{2}}+2ax+cy+a=0\] and \[{{x}^{2}}+{{y}^{2}}-3ax+dy-1=0\] intersect in two distinct points \[P\] and \[Q\] then the line \[5x+by-a=0\] passes through \[P\] and \[Q\]  for                                             [AIEEE 2005]

    A)             Infinitely many values of \[a\]  

    B)             Exactly two values of \[a\]

    C)             Exactly one value of \[a\]         

    D)          No value of \[a\]

    Correct Answer: D

    Solution :

     Equation of line PQ (i.e., common chord) is \[5ax+(c-d)y+a+1=0\]                            .....(i)            Also given equation of line PQ is            \[5x+by-a=0\]                              ..... (ii)            Therefore \[\frac{5a}{5}=\frac{c-d}{b}=\frac{a+1}{-a};\] As \[\frac{a+1}{-a}=a\]            Þ \[{{a}^{2}}+a+1=0\]                     Therefore no real value of a exists, (as D<0).


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