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JEE Main & Advanced Mathematics Complex Numbers and Quadratic Equations Question Bank Relation between roots and coefficients

  • question_answer
    The roots of the equation \[{{x}^{2}}+ax+b=0\]are p,  and q, then the equation whose roots are \[{{p}^{2}}q\] and \[p{{q}^{2}}\] will be [MP PET 1980]

    A) \[{{x}^{2}}+abx+{{b}^{3}}=0\]

    B) \[{{x}^{2}}-abx+{{b}^{3}}=0\]

    C) \[b{{x}^{2}}+x+a=0\]

    D) \[{{x}^{2}}+ax+ab=0\]

    Correct Answer: A

    Solution :

    \[f(x)={{x}^{2}}+ax+b=0\]Þ \[p+q=-a\]and\[pq=b\] Now required equation whose roots are \[{{p}^{2}}q\]and \[p{{q}^{2}}\] Therefore sum of roots \[={{p}^{2}}q+p{{q}^{2}}=pq(p+q)=-ab\]  and product of roots =\[p{{q}^{2}}.q{{p}^{2}}={{(pq)}^{3}}={{b}^{3}}\] Thus equation is\[{{x}^{2}}+abx+{{b}^{3}}=0\].


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