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Question

If b is the mean proportion between a & c,show that b(a+c)is the mean proportion between (a*a+b*b)&(b*b+c*c)


Solution

Correct question  : If b is the mean proportional between a and c, prove that  ( ab + bc ) is mean proportional of (a 2+b 2) and (b 2+c 2) .

Solution  :  Given  : b is the mean proportional between a and c , So 

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 )

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