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  • If the roots of the equation 1/x+a +1/x+b=1/c are equal in magnitude but opposite in sign , then their product is

If the roots of the equation 1/x+a +1/x+b=1/c are equal in magnitude but opposite in sign , then their product is

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Solution:    Given,   

                1/x+a +1/x+b =1/c

         Rightarrow   c(x+b+x+a)=(x+a)(x+b)

          Rightarrow   x^2+(a+b-2c)x+ab-ac-bc=0

         Let  alpha and -alpha be the roots of the equation

                 0=alpha +(-alpha )=a+b-2c   Rightarrow   c=1/2 (a+b)

       Also ,          2alpha (-alpha )=2ab-2(a+b)c=2ab-(a+b)^2=-(a^2+b^2)

          Rightarrow        alpha (-alpha )=-1/2 (a^2+b^2)

     

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Deependra Verma

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