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JEE Main & Advanced Mathematics Straight Line Question Bank Concurrency of three lines

  • question_answer
    The lines             \[(p-q)x+(q-r)y+(r-p)=0\]                              \[(q-r)x+(r-p)y+(p-q)=0\]            \[(r-p)x+(p-q)y+(q-r)=0\]are

    A)            Parallel                                      

    B)            Perpendicular

    C)            Concurrent                                

    D)            None of these

    Correct Answer: C

    Solution :

               \[\left| \,\begin{matrix}    p-q & q-r & r-p  \\    q-r & r-p & p-q  \\    r-p & p-q & q-r  \\ \end{matrix}\, \right|=\left| \,\begin{matrix}    0 & q-r & r-p  \\    0 & r-p & p-q  \\    0 & p-q & q-r  \\ \end{matrix}\, \right|=0\]                    Hence the lines are concurrent.               Aliter: Since sum of the coefficient of x, y and the constant term is zero, hence the lines are concurrent.


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