InterviewSolution
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InterviewSolution
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If `A_(1),A_(2),A_(3),………………,A_(n),a_(1),a_(2),a_(3),……a_(n),a,b,c epsilonR` show that the roots of the equation `(A_(1)^(2))/(x-a_(1))+(A_(2)^(2))/(x-a_(2))+(A_(3)^(2))/(x-a_(3))+…+(A_(n)^(2))/(x-a_(n))` `=ab^(2)+c^(2) x+ac `are real. |
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Answer» Assume `alpha + i beta` is a complex root of the given equation, then conjugate of this root i.e. `alpha-ibeta` is also root of this equation. On putting `x=alpha+ibeta` and `x=alpha-ibeta` in the given equation we get `(A_(1)^(2))/(alpha+ibeta-a_(1))+(A_(2)^(2))/(alpah+i beta-a_(2))+(A_(3)^(2))/(alpha+i beta-a_(3))+....+(A_(n)^(2))/(alpha+ibeta-a_(n))` `=ab^(2)+c^(2)(alpha+ibeta)+ac`...i and `(A_(1)^(2))/(alpha -ibeta-a_(1))+(A_(2)^(2))/(alpha-ibeta-a_(2))+(A_(3)^(2))/(alpha-ibeta-a_(3))+..........+(A_(n)^(2))/(alpha-i beta-a_(n))` `=ab^(2)+c^(2)(alpha-i beta)+ac` ...........ii On subtracting Eq. i from Eq ii we get `2i beta[(A_(1)^(2))/((alpha-a_(1)^(2)+beta^(2))+(A_(2)^(2))/((alpha-a_(2))^(2)+beta^(2))+(A_(3)^(2))/((alpha-a_(3))^(2)+beta^(2))` `+...........+(A_(n)^(2))/((alpha-a_(n))^(2)+beta^(2))+c^(2)]=0` The expression in bracket `!=0` `:.2ibeta=0impliesbeta=0` Hence all roots of the given equation are real. |
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