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

Solution:It is given that

y is mean proportional between x and z

We can write it as

y

^{2}= xz …… (1)Consider

LHS = xyz (x + y + z)

^{3}It can be written as

= xz. y (x + y + z)

^{3}Using equation (1)

= y

^{2}y (x + y + z)^{3}= y

^{3}(x + y + z)^{3}So we get

= [y (x + y + z)]

^{3}By further calculation

= (xy + y

^{2}+ yz)^{3}Using equation (1)

= (xy + yz + zx)

^{3}= RHS

Hence, it is proved.