InterviewSolution
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InterviewSolution
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If `alpha,beta`are roots of `x^2+-p x+1=0a n dgamma,delta`are the roots of `x^2+q x+1=0`, then prove that `q^2-p^2=(alpha-gamma)(beta-gamma)(alpha+delta)(beta+delta)`. |
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Answer» Correct Answer - `q^(2)-p^(2)` Since, `alpha+beta=-, alphabeta=1and gamma+delta=-q,gamma delta=1` Now, `(alpha-gamma)(beta-gamma)(alpha+delta)(beta+delta)` `{alpha beta-gamma(alpha+beta)+gamma^(2)}{alphabeta+delta(alpha+beta)+delta^(2)}` `={1-gamma(-p)+gamma^(2)}{1+delta(-p)+delta^(2)}` `={1-gamma(-p)+gamma^(2)}{1+delta(-p)+delta^(2)}` `=(1+y^(2)+gammap)(1-deltap+delta^(2))=(-qgamma+gammap)(-deltap-deltap)` `[becausegamma^(2)+qgamma+1=0and delta^(2)+qdelta+1=0]` `=(q^(2)-p^(2))(gammadelta)=q^(2)-p^(2)" "[becausegamma delta=1]` |
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