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If y is mean proportional between x and z, prove that xyz (x + y + z)3 = (xy + yz + zx)3.

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exam oriented an important question from ML aggarwal, class 10th, chapter 7, ratio and proportion, Avichal publication

It is given that y is mean proportional between x and z,  and we have to prove that xyz (x + y + z)3 = (xy + yz + zx)3.

Question 14, exercise 7.2

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  1. Solution:

    It is given that

    y is mean proportional between x and z

    We can write it as

    y2 = xz …… (1)

    Consider

    LHS = xyz (x + y + z)3

    It can be written as

    = xz. y (x + y + z)3

    Using equation (1)

    = y2 y (x + y + z)3

    = y3 (x + y + z)3

    So we get

    = [y (x + y + z)]3

    By further calculation

    = (xy + y2 + yz)3

    Using equation (1)

    = (xy + yz + zx)3

    = RHS

    Hence, it is proved.

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