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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  the two equations \[{{x}^{2}}-cx+d=0\] and \[{{x}^{2}}-ax+b=0\] have one common root and the second has equal roots, then \[2(b+d)=\]

    A) 0

    B) \[a+c\]

    C) \[ac\]

    D) \[-ac\]

    Correct Answer: C

    Solution :

    Let roots of \[{{x}^{2}}-cx+d=0\]be \[\alpha ,\beta \] then roots of \[{{x}^{2}}-ax+b=0\]be \[\alpha ,\alpha \] \ \[\alpha +\beta =c,\alpha \beta =d,\alpha +\alpha =a,{{\alpha }^{2}}=b\] Hence \[2(b+d)=2({{\alpha }^{2}}+\alpha \beta )=2\alpha (\alpha +\beta )=ac\]


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