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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 \[{{x}^{2}}+ax+10=0\] and \[{{x}^{2}}+bx-10=0\] have a common root, then \[{{a}^{2}}-{{b}^{2}}\] is equal to  [Kerala (Engg.) 2002]

    A) 10

    B) 20

    C) 30

    D) 40

    Correct Answer: D

    Solution :

    Let a be a common root, then          \[{{\alpha }^{2}}+a\alpha +10=0\]                        ?..(i) and  \[{{\alpha }^{2}}+b\alpha -10=0\]                        ?..(ii) form (i) - (ii), \[(a-b)\alpha +20=0\Rightarrow \alpha =-\frac{20}{a-b}\] Substituting the value of a in (i), we get \[{{\left( -\frac{20}{a-b} \right)}^{2}}+a\,\left( -\frac{20}{a-b} \right)+10=0\] \[\Rightarrow 400-20\,a(a-b)+10{{(a-b)}^{2}}=0\] \[\Rightarrow 40-2{{a}^{2}}+2ab+{{a}^{2}}+{{b}^{2}}-2ab=0\] \[\Rightarrow \,{{a}^{2}}-{{b}^{2}}=40\].


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