Question

if b is the mean proportional of a and c prove that the mean proportional of a2+b2 and b2+c2 is ab+ bc

Answer 1

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

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:)