Question no.13 From ML aggarwal book, class10, chapter 7, ratio and proportion
In this ques it is given that if b is the mean proportional between a and c, then we have to prove that (ab + bc) is the mean proportional between (a2 + b2) and (b2 + c2).
Question 13, exercise 7.2
Solution:
It is given that
b is the mean proportional between a and c
b2 = ac …. (1)
Here (ab + bc) is the mean proportional between (a2 + b2) and (b2 + c2)
(ab + bc)2 = (a2 + b2) (b2 + c2)
Consider LHS = (ab + bc)2
Expanding using formula
= a2b2 + b2c2 + 2ab2c
Using equation (1)
= a2 (ac) + ac (c)2 + 2a. ac. c
= a3c + ac3 + 2a2c2
Taking ac as common
= ac (a2 + c2 + 2ac)
= ac (a + c)2
RHS = (a2 + b2) (b2 + c2)
Using equation (1)
= (a2 + ac) (ac + c2)
Taking common terms out
= a (a + c) c (a + c)
= ac (a + c)2
Hence, LHS = RHS.