11th Class Mathematics Complex Numbers and Quadratic Equations Question Bank Critical Thinking

  • question_answer
    If \[\sin \alpha ,\cos \alpha \] are the roots of the equation \[a{{x}^{2}}+bx+c=0\], then [MP PET 1993]

    A) \[{{a}^{2}}-{{b}^{2}}+2ac=0\]

    B) \[{{(a-c)}^{2}}={{b}^{2}}+{{c}^{2}}\]

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

    D) \[{{a}^{2}}+{{b}^{2}}+2ac=0\]

    Correct Answer: A

    Solution :

    As given, \[\sin \alpha +\cos \alpha =-\frac{b}{a},\]\[\sin \alpha \cos \alpha =\frac{c}{a}\] To eliminate\[\alpha \], we have \[1={{\sin }^{2}}\alpha +{{\cos }^{2}}\alpha ={{(\sin \alpha +\cos \alpha )}^{2}}-2\sin \alpha \cos \alpha \]     \[=\frac{{{b}^{2}}}{{{a}^{2}}}-\frac{2c}{a}\,\,\Rightarrow {{a}^{2}}-{{b}^{2}}+2ac=0\]

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