Question

If b is the mean Proportional between a and c . prove that (ab+bc) is the mean Proportional between (a^2+b^2) and (b^2+c^2)

Answer 1

Answer:

Proved below.

Step-by-step explanation:

Given,

     b is the mean proportional between a and c.

It means :

        a / b = b / c

  ⇒ ac = b^2       ...( 1 )

Now,

⇒ b^2( a + c )^2 = b^2( a + c )^2

   ⇒ ac( a + c )^2           { from ( 1 ) }

   ⇒ a( a + c )c( a + c )

   ⇒ ( a^2 + ac )( ac + c^2 )

   ⇒ ( a^2 + b^2 )( b^2 + c^2 )

⇒ b^2( a + c )^2 = ( a^2 + b^2 )( b^2 + c^2 )

⇒ ( ab + ac )^2 = ( a^2 + b^2 )( b^2 + c^2 )

⇒ ( ab + ac ) / ( b^2 + c^2 ) = ( a^2 + b^2 ) / ( ab + ac )

 The above relation says that ab + bc is the mean proportional between a^2 + b^2 and b^2 + c^2.