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

  • question_answer If \[a,b,c\] are in G.P., then the equations \[a{{x}^{2}}+2bx+c=0\] and \[d{{x}^{2}}+2ex+f=0\] have a common root if \[\frac{d}{a},\frac{e}{b},\frac{f}{c}\] are in [IIT 1985; Pb. CET 2000; DCE 2000]

    A) A.P.

    B) G.P.

    C) H.P.

    D) None of these

    Correct Answer: A

    Solution :

    As given, \[{{b}^{2}}=ac\]Þ equation \[a{{x}^{2}}+2bx+c=0\]can be written as \[a{{x}^{2}}+2\sqrt{ac}x+c=0\] Þ \[{{(\sqrt{a}x+\sqrt{c})}^{2}}=0\] Þ \[x=-\sqrt{\frac{c}{a}}\] (repeated root) This must be the common root by hypothesis. So it must satisfy the equation \[d{{x}^{2}}+2ex+f=0\] Þ \[d\frac{c}{a}-2e\sqrt{\frac{c}{a}}+f=0\] Þ \[\frac{d}{a}+\frac{f}{c}=\frac{2e}{c}\sqrt{\frac{c}{a}}=\frac{2e}{b}\] Þ \[\frac{d}{a},\frac{e}{b},\frac{f}{c}\]are in A.P.

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