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Question

If b is the mean proportional between a and c, prove that (ab+bc) is the mean proportional between (a2+b2) and (b2+c2).

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Solution

It is given that

b is the mean proportional that is geometric mean between a and c.

b2=ac(1)

And we have to prove (ab+bc) is the mean proportional between (a2+b2) and (b2+c2) that is we have to prove,

(ab+bc)2=(a2+b2)(b2+c2)

Consider LHS=(ab+bc)2

Expanding using formula

LHS=a2b2+b2c2+2ab2c

Using equation (1)

LHS=a2(ac)+ac(c)2+2a.ac.c

=a3c+ac3+2a2c2

Taking ac as common

LHS=ac(a2+c2+2ac)

LHS=ac(a+c)2(2)

Now,

RHS=(a2+b2)(b2+c2)

Using equation (1)

RHS=(a2+ac)(ac+c2)

Taking common terms out

RHS=a(a+c)c(a+c)

RHS=ac(a+c)2(3)

From (2) and (3)

LHS=RHS

Therefore it is proved that (ab+bc) is the mean proportional between (a2+b2) and (b2+c2).

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