Question

If b is the mean proportion between a and c, prove that a, c, a2+b2 and b2+c2 are proportional

Answer 1

Answer:

Solution : -

here it is clearly given that b is the mean proportional between a and c .

therefore, b2 = ac

Now (a 2+b 2)(b 2+c 2) = (a 2+ ac )( ac +c 2) ,

(a 2+b 2)(b 2+c 2) = a(a + c ) c( a + c ) ,

(a 2+b 2)(b 2+c 2) = a c (a + c )2 ,

(a 2+b 2)(b 2+c 2) = b2 (a + c )2

(a 2+b 2)(b 2+c 2) = (ab + bc )2

So,

( ab + bc ) is mean proportional of (a 2+b 2) and (b 2+c 2) . ( Hence proved )

Answer 2

Answer:

a and c, prove that a,c,a2+b2 and b2+c2 are proportional.

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It is given that

b is the mean proportional between a and c

We can write it as

b2=a×c

b2=ac⋯⋯(1)

We have to prove that a,c,a2+b2 and b2+c2 are in proportion

That is we have to prove 

ca=(b2+c2)(a2+b2)

Consider, RHS=(b2+c2)(a2+b2)

=ac+c2a2+ac⋯⋯[from (1)]

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

=ca=LHS