Question

if b is the mean proportion between a and c show that â4+â2b'2+b b'4/b'4+b'2č2+č4=â2/č2

Answer 1

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 )