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
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.