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CEE Kerala Engineering CEE Kerala Engineering Solved Paper-2001

  • question_answer
    If the expression \[a(b-c){{x}^{2}}+b(c-a)xy+c(a-b){{y}^{2}}\]is a perfect square, then a, b, c are in:

    A)  AP                                        

    B)  HP

    C)  GP                        

    D)         both AP and GP

    E)  none of these

    Correct Answer: B

    Solution :

    Given expression is \[a(b-c){{x}^{2}}+b(c-a)xy+c(a-b){{y}^{2}}\] Put\[x=1\]in the given expression \[\therefore \]  \[a(b-c)+b(c-a)+c(a-b)\] \[=ab-ac+bc-ab+ca-cb\] \[=0\] Since, it is a perfect square, therefore both roots are same i. e., 1,1. \[\therefore \]Sum of roots \[=-\frac{b(c-a)}{a(b-c)}\] \[\Rightarrow \]               \[1+1=\frac{-bc+ab}{ab-ac}\] \[\Rightarrow \]               \[2ab-2ac=-bc+ab\] \[\Rightarrow \]               \[ab+bc=2ac\] \[\Rightarrow \]               \[b(a+c)=2ac\] \[\Rightarrow \]               \[b=\frac{2ac}{a+c}\] Hence, a, b, c are in HP.


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