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JEE Main & Advanced Mathematics Complex Numbers and Quadratic Equations Question Bank Condition for common roots, Quadratic expressions and Position of roots

  • question_answer
    If every pair of the equations\[{{x}^{2}}+px+qr=0\], \[{{x}^{2}}+qx+rp=0,\] \[{{x}^{2}}+rx+pq=0\] have a common root, then the sum of three common roots is

    A) \[\frac{-(p+q+r)}{2}\]

    B) \[\frac{-p+q+r}{2}\]

    C) \[-(p+q+r)\]

    D) \[-p+q+r\]

    Correct Answer: A

    Solution :

    Let the roots be \[\alpha ,\beta ;\beta ,\gamma \]and \[\gamma ,\alpha \]respectively. \[\therefore \,\,\,\,\,\alpha +\beta =-p,\ \beta +\gamma =-q,\gamma +\alpha =-r\] Adding all, we get\[\Sigma \alpha =-(p+q+r)/2\]etc.


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